Section 1.6
             Limits involving Infinity

                     V63.0121.021, Calculus I

                       ...
Announcements




         Quiz 1 is next week in
         recitation. Covers Sections
         1.1–1.4




              ...
Objectives




         “Intuit” limits involving
         infinity by eyeballing the
         expression.
         Show l...
Recall the definition of limit



Definition
We write
                                         lim f(x) = L
              ...
Recall the unboundedness problem
                          1
Recall why lim+             doesn’t exist.
                  ...
Recall the unboundedness problem
                          1
Recall why lim+             doesn’t exist.
                  ...
Recall the unboundedness problem
                          1
Recall why lim+             doesn’t exist.
                  ...
Recall the unboundedness problem
                          1
Recall why lim+             doesn’t exist.
                  ...
Outline



Infinite Limits
    Vertical Asymptotes
    Infinite Limits we Know
    Limit “Laws” with Infinite Limits
    I...
Infinite Limits


Definition
The notation
                                                                    y
          ...
Infinite Limits


Definition
The notation
                                                                    y
          ...
Infinite Limits


Definition
The notation
                                                                    y
          ...
Infinite Limits


Definition
The notation
                                                                    y
          ...
Infinite Limits


Definition
The notation
                                                                    y
          ...
Infinite Limits


Definition
The notation
                                                                    y
          ...
Infinite Limits


Definition
The notation
                                                                    y
          ...
Infinite Limits


Definition
The notation
                                                                    y
          ...
Negative Infinity



Definition
The notation
                                       lim f(x) = −∞
                        ...
Negative Infinity



Definition
The notation
                                       lim f(x) = −∞
                        ...
Vertical Asymptotes



Definition
The line x = a is called a vertical asymptote of the curve y = f(x) if at
least one of t...
Infinite Limits we Know

                                                                          y
                     ...
Infinite Limits we Know

                                                                          y
                     ...
Infinite Limits we Know

                                                                          y
                     ...
Finding limits at trouble spots



Example
Let
                                                    x2 + 2
                ...
Finding limits at trouble spots



Example
Let
                                                    x2 + 2
                ...
Use the number line



                .                                                                          . x − 1)...
Use the number line


             −
             ..                       0
                                      ..     ...
Use the number line


             −
             ..                       0
                                      ..     ...
Use the number line


             −
             ..                       0
                                      ..     ...
Use the number line


             −
             ..                       0
                                      ..     ...
Use the number line


             −
             ..                       0
                                      ..     ...
Use the number line




                                    . mall
             −
             ..                       0
...
Use the number line




                                   . mall
             −
             ..                      0
  ...
Use the number line


             −
             ..                      0
                                     ..       ...
Use the number line


             −
             ..                      0
                                     ..       ...
Use the number line


             −
             ..                      0
                                     ..       ...
Use the number line


             −
             ..                      0
                                     ..       ...
In English, now




To explain the limit, you can say:
“As x → 1− , the numerator approaches 3, and the denominator
approa...
The graph so far

                              lim f(x) = + ∞                  lim f(x) = − ∞
                           ...
The graph so far

                              lim f(x) = + ∞                  lim f(x) = − ∞
                           ...
The graph so far

                              lim f(x) = + ∞                  lim f(x) = − ∞
                           ...
The graph so far

                              lim f(x) = + ∞                  lim f(x) = − ∞
                           ...
The graph so far

                              lim f(x) = + ∞                  lim f(x) = − ∞
                           ...
Limit Laws (?) with infinite limits



Fact
The sum of two positive or two negative infinite limits is infinite.
      If ...
Rules of Thumb with infinite limits

                                                                     ∞
              ...
Rules of Thumb with infinite limits

                                                                     ∞
              ...
Rules of Thumb with infinite limits

                                                                     ∞
              ...
Rules of Thumb with infinite limits
Kids, don't try this at home!




Fact
The sum of a finite limit and an infinite limit...
Rules of Thumb with infinite limits
Kids, don't try this at home!




Fact
The sum of a finite limit and an infinite limit...
Rules of Thumb with infinite limits
Kids, don't try this at home!




Fact
The product of a finite limit and an infinite l...
Rules of Thumb with infinite limits
Kids, don't try this at home!
                                         {
             ...
Rules of Thumb with infinite limits
Kids, don't try this at home!
                                           {
           ...
Multiplying infinite limits
Kids, don't try this at home!




Fact
The product of two infinite limits is infinite.
       ...
Multiplying infinite limits
Kids, don't try this at home!
                                                                ...
Dividing by Infinity
Kids, don't try this at home!




Fact
The quotient of a finite limit by an infinite limit is zero.
 ...
Dividing by Infinity
Kids, don't try this at home!




Fact
The quotient of a finite limit by an infinite limit is zero.
 ...
Dividing by zero is still not allowed




                                     1 .
                                     . ...
Indeterminate Limit forms

                   L
Limits of the form   are indeterminate. There is no rule for evaluating
  ...
Indeterminate Limit forms


Limits of the form 0 · ∞ and ∞ − ∞ are also indeterminate.
Example
                           ...
Indeterminate forms are like Tug Of War




Which side wins depends on which side is stronger.
                           ...
Outline



Infinite Limits
    Vertical Asymptotes
    Infinite Limits we Know
    Limit “Laws” with Infinite Limits
    I...
Definition
Let f be a function defined on some interval (a, ∞). Then

                                         lim f(x) = ...
Definition
Let f be a function defined on some interval (a, ∞). Then

                                         lim f(x) = ...
Definition
Let f be a function defined on some interval (a, ∞). Then

                                         lim f(x) = ...
Basic limits at infinity




Theorem
Let n be a positive integer. Then
           1
      lim     =0
     x→∞ xn
         ...
Limit laws at infinity


Fact
Any limit law that concerns finite limits at a finite point a is still true if
the finite po...
Using the limit laws to compute limits at ∞

Example
                   x
Find lim
       x→∞ x2      +1




             ...
Using the limit laws to compute limits at ∞

Example
                   x
Find lim
       x→∞ x2      +1

Answer
The limit...
Solution

Solution
Factor out the largest power of x from the numerator and denominator.
We have
                     x   ...
Using the limit laws to compute limits at ∞

Example
                   x
Find lim
       x→∞ x2      +1

Answer
The limit...
Solution

Solution
Factor out the largest power of x from the numerator and denominator.
We have
                     x   ...
Another Example



Example
Find
                                      2x3 + 3x + 1
                                   lim
...
Another Example



Example
Find
                                      2x3 + 3x + 1
                                   lim
...
Solution

Solution
Factor out the largest power of x from the numerator and denominator.
We have
                         ...
Solution

Solution
Factor out the largest power of x from the numerator and denominator.
We have
                         ...
Still Another Example



Example
Find                                      √
                                           3x...
Still Another Example

                                                              √          √     √
                  ...
Solution


Solution

                                √              √
                                 3x4 + 7         x4 ...
Rationalizing to get a limit
Example
                       (√                 )
Compute lim                  4x2 + 17 − 2...
Rationalizing to get a limit
Example
                       (√                 )
Compute lim                  4x2 + 17 − 2...
Kick it up a notch
Example
                       (√                  )
Compute lim                  4x2 + 17x − 2x .
    ...
Kick it up a notch
Example
                       (√                  )
Compute lim                  4x2 + 17x − 2x .
    ...
Summary




     Infinity is a more complicated concept than a single number.
     There are rules of thumb, but there are...
Upcoming SlideShare
Loading in …5
×

Lesson 6: Limits Involving ∞ (Section 21 slides)

829 views

Published on

0 Comments
1 Like
Statistics
Notes
  • Be the first to comment

No Downloads
Views
Total views
829
On SlideShare
0
From Embeds
0
Number of Embeds
4
Actions
Shares
0
Downloads
32
Comments
0
Likes
1
Embeds 0
No embeds

No notes for slide

Lesson 6: Limits Involving ∞ (Section 21 slides)

  1. 1. Section 1.6 Limits involving Infinity V63.0121.021, Calculus I New York University September 23, 2010 Announcements Quiz 1 is next week in recitation. Covers Sections 1.1–1.4 . . . . . .
  2. 2. Announcements Quiz 1 is next week in recitation. Covers Sections 1.1–1.4 . . . . . . V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 2 / 38
  3. 3. Objectives “Intuit” limits involving infinity by eyeballing the expression. Show limits involving infinity by algebraic manipulation and conceptual argument. . . . . . . V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 3 / 38
  4. 4. Recall the definition of limit Definition We write lim f(x) = L x→a and say “the limit of f(x), as x approaches a, equals L” if we can make the values of f(x) arbitrarily close to L (as close to L as we like) by taking x to be sufficiently close to a (on either side of a) but not equal to a. . . . . . . V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 4 / 38
  5. 5. Recall the unboundedness problem 1 Recall why lim+ doesn’t exist. x→0 x y . .?. L . x . No matter how thin we draw the strip to the right of x = 0, we cannot “capture” the graph inside the box. . . . . . . V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 5 / 38
  6. 6. Recall the unboundedness problem 1 Recall why lim+ doesn’t exist. x→0 x y . .?. L . x . No matter how thin we draw the strip to the right of x = 0, we cannot “capture” the graph inside the box. . . . . . . V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 5 / 38
  7. 7. Recall the unboundedness problem 1 Recall why lim+ doesn’t exist. x→0 x y . .?. L . x . No matter how thin we draw the strip to the right of x = 0, we cannot “capture” the graph inside the box. . . . . . . V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 5 / 38
  8. 8. Recall the unboundedness problem 1 Recall why lim+ doesn’t exist. x→0 x y . .?. L . x . No matter how thin we draw the strip to the right of x = 0, we cannot “capture” the graph inside the box. . . . . . . V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 5 / 38
  9. 9. Outline Infinite Limits Vertical Asymptotes Infinite Limits we Know Limit “Laws” with Infinite Limits Indeterminate Limit forms Limits at ∞ Algebraic rates of growth Rationalizing to get a limit . . . . . . V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 6 / 38
  10. 10. Infinite Limits Definition The notation y . lim f(x) = ∞ x→a means that values of f(x) can be made arbitrarily large (as large as we please) by taking x sufficiently close to a but not equal to a. . x . . . . . . . V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 7 / 38
  11. 11. Infinite Limits Definition The notation y . lim f(x) = ∞ x→a means that values of f(x) can be made arbitrarily large (as large as we please) by taking x sufficiently close to a but not equal to a. . x . “Large” takes the place of “close to L”. . . . . . . V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 7 / 38
  12. 12. Infinite Limits Definition The notation y . lim f(x) = ∞ x→a means that values of f(x) can be made arbitrarily large (as large as we please) by taking x sufficiently close to a but not equal to a. . x . “Large” takes the place of “close to L”. . . . . . . V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 7 / 38
  13. 13. Infinite Limits Definition The notation y . lim f(x) = ∞ x→a means that values of f(x) can be made arbitrarily large (as large as we please) by taking x sufficiently close to a but not equal to a. . x . “Large” takes the place of “close to L”. . . . . . . V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 7 / 38
  14. 14. Infinite Limits Definition The notation y . lim f(x) = ∞ x→a means that values of f(x) can be made arbitrarily large (as large as we please) by taking x sufficiently close to a but not equal to a. . x . “Large” takes the place of “close to L”. . . . . . . V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 7 / 38
  15. 15. Infinite Limits Definition The notation y . lim f(x) = ∞ x→a means that values of f(x) can be made arbitrarily large (as large as we please) by taking x sufficiently close to a but not equal to a. . x . “Large” takes the place of “close to L”. . . . . . . V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 7 / 38
  16. 16. Infinite Limits Definition The notation y . lim f(x) = ∞ x→a means that values of f(x) can be made arbitrarily large (as large as we please) by taking x sufficiently close to a but not equal to a. . x . “Large” takes the place of “close to L”. . . . . . . V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 7 / 38
  17. 17. Infinite Limits Definition The notation y . lim f(x) = ∞ x→a means that values of f(x) can be made arbitrarily large (as large as we please) by taking x sufficiently close to a but not equal to a. . x . “Large” takes the place of “close to L”. . . . . . . V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 7 / 38
  18. 18. Negative Infinity Definition The notation lim f(x) = −∞ x→a means that the values of f(x) can be made arbitrarily large negative (as large as we please) by taking x sufficiently close to a but not equal to a. . . . . . . V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 8 / 38
  19. 19. Negative Infinity Definition The notation lim f(x) = −∞ x→a means that the values of f(x) can be made arbitrarily large negative (as large as we please) by taking x sufficiently close to a but not equal to a. We call a number large or small based on its absolute value. So −1, 000, 000 is a large (negative) number. . . . . . . V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 8 / 38
  20. 20. Vertical Asymptotes Definition The line x = a is called a vertical asymptote of the curve y = f(x) if at least one of the following is true: lim f(x) = ∞ lim f(x) = −∞ x→a x→a lim f(x) = ∞ lim f(x) = −∞ x→a+ x→a+ lim f(x) = ∞ lim f(x) = −∞ x→a− x→a− . . . . . . V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 9 / 38
  21. 21. Infinite Limits we Know y . . . 1 lim+ = ∞ x→0 x . . . . . . . . x . . . . . . . . . . V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 10 / 38
  22. 22. Infinite Limits we Know y . . . 1 lim+ = ∞ x→0 x . 1 lim = −∞ x→0− x . . . . . . . x . . . . . . . . . . V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 10 / 38
  23. 23. Infinite Limits we Know y . . . 1 lim+ = ∞ x→0 x . 1 lim = −∞ x→0− x . . . . . . . x . 1 lim =∞ x→0 x2 . . . . . . . . . V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 10 / 38
  24. 24. Finding limits at trouble spots Example Let x2 + 2 f(x) = x2 − 3x + 2 Find lim f(x) and lim+ f(x) for each a at which f is not continuous. x→a− x→a . . . . . . V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 11 / 38
  25. 25. Finding limits at trouble spots Example Let x2 + 2 f(x) = x2 − 3x + 2 Find lim f(x) and lim+ f(x) for each a at which f is not continuous. x→a− x→a Solution The denominator factors as (x − 1)(x − 2). We can record the signs of the factors on the number line. . . . . . . V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 11 / 38
  26. 26. Use the number line . . x − 1) ( . . . . . . . . . . . . . V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 12 / 38
  27. 27. Use the number line − .. 0 .. . + . x − 1) . . ( 1 . . . . . . . . . . . . . . V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 12 / 38
  28. 28. Use the number line − .. 0 .. . + . x − 1) . . ( 1 . − . 0 .. . + . x − 2) . . ( 2 . . . . . . . . . . . . . . V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 12 / 38
  29. 29. Use the number line − .. 0 .. . + . x − 1) . . ( 1 . − . 0 .. . + . x − 2) . . ( 2 . . + . x2 + 2) ( . . . . . . . . . . . . . V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 12 / 38
  30. 30. Use the number line − .. 0 .. . + . x − 1) . . ( 1 . − . 0 .. . + . x − 2) . . ( 2 . . + . x2 + 2) ( . . .. . . .. . f .(x) 1 . 2 . . . . . . . V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 12 / 38
  31. 31. Use the number line − .. 0 .. . + . x − 1) . . ( 1 . − . 0 .. . + . x − 2) . . ( 2 . . + . x2 + 2) ( .+ . .. . . .. . f .(x) 1 . 2 . . . . . . . V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 12 / 38
  32. 32. Use the number line . mall − .. 0 .. . + s . x − 1) . ( 1 . − . 0 .. . + . x − 2) . . ( 2 . . + . x2 + 2) ( .+ +∞ . . . . . .. . f .(x) 1 . 2 . lim f(x) = + ∞ x→1− . . . . . . V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 12 / 38
  33. 33. Use the number line . mall − .. 0 .. . + s . x − 1) . ( 1 . − . 0 .. . + . x − 2) . . ( 2 . . + . x2 + 2) ( .+ +∞ . . ∞ . . − . .. . f .(x) 1 . 2 . lim f(x) = + ∞ x→1− lim f(x) = − ∞ x→1+ . . . . . . V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 12 / 38
  34. 34. Use the number line − .. 0 .. . + . x − 1) . . ( 1 . − . 0 .. . + . x − 2) . . ( 2 . . + . x2 + 2) ( .+ +∞ . . ∞ . . . − − .. . f .(x) 1 . 2 . lim f(x) = + ∞ x→1− lim f(x) = − ∞ x→1+ . . . . . . V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 12 / 38
  35. 35. Use the number line − .. 0 .. . + . x − 1) . . ( small 1 . − . 0 .. . + . x − 2) . . ( 2 . . + . x2 + 2) ( .+ +∞ . . ∞ . . ∞ . . . − − − . f .(x) 1 . 2 . lim f(x) = + ∞ lim f(x) = − ∞ x→1− x→2− lim f(x) = − ∞ x→1+ . . . . . . V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 12 / 38
  36. 36. Use the number line − .. 0 .. . + . x − 1) . . ( small 1 . − . 0 .. . + . x − 2) . . ( 2 . . + . x2 + 2) ( .+ +∞ . . ∞ . . ∞ . . ∞ . − − − + . f .(x) 1 . 2 . lim f(x) = + ∞ lim f(x) = − ∞ x→1− x→2− lim f(x) = − ∞ lim f(x) = + ∞ x→1+ x→2+ . . . . . . V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 12 / 38
  37. 37. Use the number line − .. 0 .. . + . x − 1) . . ( 1 . − . 0 .. . + . x − 2) . . ( 2 . . + . x2 + 2) ( .+ +∞ . . ∞ . . ∞ . . ∞ . − − − + . + f .(x) 1 . 2 . lim f(x) = + ∞ lim f(x) = − ∞ x→1− x→2− lim f(x) = − ∞ lim f(x) = + ∞ x→1+ x→2+ . . . . . . V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 12 / 38
  38. 38. In English, now To explain the limit, you can say: “As x → 1− , the numerator approaches 3, and the denominator approaches 0 while remaining positive. So the limit is +∞.” . . . . . . V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 13 / 38
  39. 39. The graph so far lim f(x) = + ∞ lim f(x) = − ∞ x→1− x→2− lim f(x) = − ∞ lim f(x) = + ∞ x→1+ x→2+ y . . . . . . . x − . 1 1 . 2 . 3 . . . . . . . V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 14 / 38
  40. 40. The graph so far lim f(x) = + ∞ lim f(x) = − ∞ x→1− x→2− lim f(x) = − ∞ lim f(x) = + ∞ x→1+ x→2+ y . . . . . . . x − . 1 1 . 2 . 3 . . . . . . . V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 14 / 38
  41. 41. The graph so far lim f(x) = + ∞ lim f(x) = − ∞ x→1− x→2− lim f(x) = − ∞ lim f(x) = + ∞ x→1+ x→2+ y . . . . . . . x − . 1 1 . 2 . 3 . . . . . . . V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 14 / 38
  42. 42. The graph so far lim f(x) = + ∞ lim f(x) = − ∞ x→1− x→2− lim f(x) = − ∞ lim f(x) = + ∞ x→1+ x→2+ y . . . . . . . x − . 1 1 . 2 . 3 . . . . . . . V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 14 / 38
  43. 43. The graph so far lim f(x) = + ∞ lim f(x) = − ∞ x→1− x→2− lim f(x) = − ∞ lim f(x) = + ∞ x→1+ x→2+ y . . . . . . . x − . 1 1 . 2 . 3 . . . . . . . V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 14 / 38
  44. 44. Limit Laws (?) with infinite limits Fact The sum of two positive or two negative infinite limits is infinite. If lim f(x) = ∞ and lim g(x) = ∞, then lim (f(x) + g(x)) = ∞. x→a x→a x→a If lim f(x) = −∞ and lim g(x) = −∞, then lim (f(x) + g(x)) = −∞. x→a x→a x→a . . . . . . V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 15 / 38
  45. 45. Rules of Thumb with infinite limits ∞ . +∞=∞ Fact The sum of two positive or two negative infinite limits is infinite. . If lim f(x) = ∞ and lim g(x) = ∞, then lim (f(x) + g(x)) = ∞. x→a x→a x→a If lim f(x) = −∞ and lim g(x) = −∞, then lim (f(x) + g(x)) = −∞. x→a x→a x→a . . . . . . V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 15 / 38
  46. 46. Rules of Thumb with infinite limits ∞ . +∞=∞ − . ∞ + (−∞) = −∞ Fact The sum of two positive or two negative infinite limits is infinite. . If lim f(x) = ∞ and lim g(x) = ∞, then lim (f(x) + g(x)) = ∞. x→a x→a x→a If lim f(x) = −∞ and lim g(x) = −∞, then lim (f(x) + g(x)) = −∞. .x→a x→a x→a . . . . . . V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 15 / 38
  47. 47. Rules of Thumb with infinite limits ∞ . +∞=∞ − . ∞ + (−∞) = −∞ Fact The sum of two positive or two negative infinite limits is infinite. . If lim f(x) = ∞ and lim g(x) = ∞, then lim (f(x) + g(x)) = ∞. x→a x→a x→a If lim f(x) = −∞ and lim g(x) = −∞, then lim (f(x) + g(x)) = −∞. .x→a x→a x→a Remark We don’t say anything here about limits of the form ∞ − ∞. . . . . . . V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 15 / 38
  48. 48. Rules of Thumb with infinite limits Kids, don't try this at home! Fact The sum of a finite limit and an infinite limit is infinite. If lim f(x) = L and lim g(x) = ±∞, then lim (f(x) + g(x)) = ±∞. x→a x→a x→a . . . . . . V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 16 / 38
  49. 49. Rules of Thumb with infinite limits Kids, don't try this at home! Fact The sum of a finite limit and an infinite limit is infinite. . If lim f(x) = L and lim g(x) = ±∞, then lim (f(x) + g(x)) = ±∞. x→a x→a x→a L+∞=∞ . L − ∞ = −∞ . . . . . . V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 16 / 38
  50. 50. Rules of Thumb with infinite limits Kids, don't try this at home! Fact The product of a finite limit and an infinite limit is infinite if the finite limit is not 0. If lim f(x) = L, lim g(x) = ∞, and L > 0, then lim f(x) · g(x) = ∞. x→a x→a x→a If lim f(x) = L, lim g(x) = ∞, and L < 0, then lim f(x) · g(x) = −∞. x→a x→a x→a If lim f(x) = L, lim g(x) = −∞, and L > 0, then x→a x→a lim f(x) · g(x) = −∞. x→a If lim f(x) = L, lim g(x) = −∞, and L < 0, then lim f(x) · g(x) = ∞. x→a x→a x→a . . . . . . V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 17 / 38
  51. 51. Rules of Thumb with infinite limits Kids, don't try this at home! { ∞ if L > 0 . ·∞= L −∞ if L < 0. Fact The product of a finite limit and an infinite limit is infinite if the finite limit is not 0. . If lim f(x) = L, lim g(x) = ∞, and L > 0, then lim f(x) · g(x) = ∞. x→a x→a x→a If lim f(x) = L, lim g(x) = ∞, and L < 0, then lim f(x) · g(x) = −∞. x→a x→a x→a If lim f(x) = L, lim g(x) = −∞, and L > 0, then x→a x→a lim f(x) · g(x) = −∞. x→a If lim f(x) = L, lim g(x) = −∞, and L < 0, then lim f(x) · g(x) = ∞. x→a x→a x→a . . . . . . V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 17 / 38
  52. 52. Rules of Thumb with infinite limits Kids, don't try this at home! { ∞ if L > 0 . ·∞= L −∞ if L < 0. Fact The product of a finite limit and an infinite limit is infinite if the finite limit is not 0. . If lim f(x) = L, lim g(x) = ∞, and L > 0, then lim f(x) · g(x) = ∞. x→a x→a x→a If lim f(x) = L, lim g(x) = ∞, and L < 0, then lim f(x) · g(x) = −∞. x→a x→a { x→a If lim f(x) =(−∞) = −∞−∞,Land0L > 0, then . · L L, lim g(x) = if > x→a x→a lim f(x) · g(x) = −∞. ∞ if L < 0. x→a If lim f(x) = L, lim g(x) = −∞, and L < 0, then lim f(x) · g(x) = ∞. x→a x→a x→a . . . . . . . V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 17 / 38
  53. 53. Multiplying infinite limits Kids, don't try this at home! Fact The product of two infinite limits is infinite. If lim f(x) = ∞ and lim g(x) = ∞, then lim f(x) · g(x) = ∞. x→a x→a x→a If lim f(x) = ∞ and lim g(x) = −∞, then lim f(x) · g(x) = −∞. x→a x→a x→a If lim f(x) = −∞ and lim g(x) = −∞, then lim f(x) · g(x) = ∞. x→a x→a x→a . . . . . . V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 18 / 38
  54. 54. Multiplying infinite limits Kids, don't try this at home! ∞·∞=∞ . ∞ · (−∞) = −∞ (−∞) · (−∞) = ∞ Fact The product of two infinite limits is infinite. . If lim f(x) = ∞ and lim g(x) = ∞, then lim f(x) · g(x) = ∞. x→a x→a x→a If lim f(x) = ∞ and lim g(x) = −∞, then lim f(x) · g(x) = −∞. x→a x→a x→a If lim f(x) = −∞ and lim g(x) = −∞, then lim f(x) · g(x) = ∞. x→a x→a x→a . . . . . . V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 18 / 38
  55. 55. Dividing by Infinity Kids, don't try this at home! Fact The quotient of a finite limit by an infinite limit is zero. f(x) If lim f(x) = L and lim g(x) = ±∞, then lim = 0. x→a x→a x→a g(x) . . . . . . V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 19 / 38
  56. 56. Dividing by Infinity Kids, don't try this at home! Fact The quotient of a finite limit by an infinite limit is zero. f(x) If lim f(x) = L and lim g(x) = ±∞, then lim = 0. x→a x→a x→a g(x) . L . =0 ∞ . . . . . . V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 19 / 38
  57. 57. Dividing by zero is still not allowed 1 . . =∞ 0 There are examples of such limit forms where the limit is ∞, −∞, undecided between the two, or truly neither. . . . . . . V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 20 / 38
  58. 58. Indeterminate Limit forms L Limits of the form are indeterminate. There is no rule for evaluating 0 such a form; the limit must be examined more closely. Consider these: 1 −1 lim =∞ lim = −∞ x→0 x2 x→0 x2 1 1 lim+ = ∞ lim = −∞ x→0 x x→0− x 1 L Worst, lim is of the form , but the limit does not exist, even x→0 x sin(1/x) 0 in the left- or right-hand sense. There are infinitely many vertical asymptotes arbitrarily close to 0! . . . . . . V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 21 / 38
  59. 59. Indeterminate Limit forms Limits of the form 0 · ∞ and ∞ − ∞ are also indeterminate. Example 1 The limit lim+ sin x · is of the form 0 · ∞, but the answer is 1. x→0 x 1 The limit lim+ sin2 x · is of the form 0 · ∞, but the answer is 0. x→0 x 1 The limit lim+ sin x · 2 is of the form 0 · ∞, but the answer is ∞. x→0 x Limits of indeterminate forms may or may not “exist.” It will depend on the context. . . . . . . V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 22 / 38
  60. 60. Indeterminate forms are like Tug Of War Which side wins depends on which side is stronger. . . . . . . V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 23 / 38
  61. 61. Outline Infinite Limits Vertical Asymptotes Infinite Limits we Know Limit “Laws” with Infinite Limits Indeterminate Limit forms Limits at ∞ Algebraic rates of growth Rationalizing to get a limit . . . . . . V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 24 / 38
  62. 62. Definition Let f be a function defined on some interval (a, ∞). Then lim f(x) = L x→∞ means that the values of f(x) can be made as close to L as we like, by taking x sufficiently large. . . . . . . V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 25 / 38
  63. 63. Definition Let f be a function defined on some interval (a, ∞). Then lim f(x) = L x→∞ means that the values of f(x) can be made as close to L as we like, by taking x sufficiently large. Definition The line y = L is a called a horizontal asymptote of the curve y = f(x) if either lim f(x) = L or lim f(x) = L. x→∞ x→−∞ . . . . . . V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 25 / 38
  64. 64. Definition Let f be a function defined on some interval (a, ∞). Then lim f(x) = L x→∞ means that the values of f(x) can be made as close to L as we like, by taking x sufficiently large. Definition The line y = L is a called a horizontal asymptote of the curve y = f(x) if either lim f(x) = L or lim f(x) = L. x→∞ x→−∞ y = L is a horizontal line! . . . . . . V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 25 / 38
  65. 65. Basic limits at infinity Theorem Let n be a positive integer. Then 1 lim =0 x→∞ xn 1 lim =0 x→−∞ xn . . . . . . V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 26 / 38
  66. 66. Limit laws at infinity Fact Any limit law that concerns finite limits at a finite point a is still true if the finite point is replaced by ±∞. That is, if lim f(x) = L and lim g(x) = M, then x→∞ x→∞ lim (f(x) + g(x)) = L + M x→∞ lim (f(x) − g(x)) = L − M x→∞ lim cf(x) = c · L (for any constant c) x→∞ lim f(x) · g(x) = L · M x→∞ f(x) L lim = (if M ̸= 0), etc. x→∞ g(x) M . . . . . . V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 27 / 38
  67. 67. Using the limit laws to compute limits at ∞ Example x Find lim x→∞ x2 +1 . . . . . . V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 28 / 38
  68. 68. Using the limit laws to compute limits at ∞ Example x Find lim x→∞ x2 +1 Answer The limit is 0. y . . x . . . . . . . V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 28 / 38
  69. 69. Solution Solution Factor out the largest power of x from the numerator and denominator. We have x x(1) 1 1 = 2 = · x2 +1 x (1 + 1/x2 ) x 1 + 1/x2 x 1 1 1 1 lim = lim = lim · lim x→∞ x2 + 1 x→∞ x 1 + 1/x2 x→∞ x x→∞ 1 + 1/x2 1 =0· = 0. 1+0 . . . . . . V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 29 / 38
  70. 70. Using the limit laws to compute limits at ∞ Example x Find lim x→∞ x2 +1 Answer The limit is 0. y . . x . Notice that the graph does cross the asymptote, which contradicts one of the commonly held beliefs of what an asymptote is. . . . . . . V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 30 / 38
  71. 71. Solution Solution Factor out the largest power of x from the numerator and denominator. We have x x(1) 1 1 = 2 = · x2 +1 x (1 + 1/x2 ) x 1 + 1/x2 x 1 1 1 1 lim = lim = lim · lim x→∞ x2 + 1 x→∞ x 1 + 1/x2 x→∞ x x→∞ 1 + 1/x2 1 =0· = 0. 1+0 Remark Had the higher power been in the numerator, the limit would have been ∞. . . . . . . V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 31 / 38
  72. 72. Another Example Example Find 2x3 + 3x + 1 lim x→∞ 4x3 + 5x2 + 7 if it exists. A does not exist B 1/2 C 0 D ∞ . . . . . . V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 32 / 38
  73. 73. Another Example Example Find 2x3 + 3x + 1 lim x→∞ 4x3 + 5x2 + 7 if it exists. A does not exist B 1/2 C 0 D ∞ . . . . . . V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 32 / 38
  74. 74. Solution Solution Factor out the largest power of x from the numerator and denominator. We have 2x3 + 3x + 1 x3 (2 + 3/x2 + 1/x3 ) = 3 4x3 + 5x2 + 7 x (4 + 5/x + 7/x3 ) 2x3 + 3x + 1 2 + 3/x2 + 1/x3 lim = lim x→∞ 4x3 + 5x2 + 7 x→∞ 4 + 5/x + 7/x3 2+0+0 1 = = 4+0+0 2 . . . . . . V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 33 / 38
  75. 75. Solution Solution Factor out the largest power of x from the numerator and denominator. We have 2x3 + 3x + 1 x3 (2 + 3/x2 + 1/x3 ) = 3 4x3 + 5x2 + 7 x (4 + 5/x + 7/x3 ) 2x3 + 3x + 1 2 + 3/x2 + 1/x3 lim = lim x→∞ 4x3 + 5x2 + 7 x→∞ 4 + 5/x + 7/x3 2+0+0 1 = = 4+0+0 2 Upshot When finding limits of algebraic expressions at infinity, look at the highest degree terms. . . . . . . V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 33 / 38
  76. 76. Still Another Example Example Find √ 3x4 + 7 lim x→∞ x2 + 3 . . . . . . V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 34 / 38
  77. 77. Still Another Example √ √ √ . 3x4 + 7 ∼ 3x4 = 3x2 Example Find √ 3x4 + 7 . lim x→∞ x2 + 3 Answer √ The limit is 3. . . . . . . V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 34 / 38
  78. 78. Solution Solution √ √ 3x4 + 7 x4 (3 + 7/x4 ) lim = lim x→∞ x2 + 3 x→∞ x2 (1 + 3/x2 ) √ x2 (3 + 7/x4 ) = lim x→∞ x2 (1 + 3/x2 ) √ (3 + 7/x4 ) = lim x→∞ 1 + 3/x2 √ 3+0 √ = = 3. 1+0 . . . . . . V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 35 / 38
  79. 79. Rationalizing to get a limit Example (√ ) Compute lim 4x2 + 17 − 2x . x→∞ . . . . . . V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 36 / 38
  80. 80. Rationalizing to get a limit Example (√ ) Compute lim 4x2 + 17 − 2x . x→∞ Solution This limit is of the form ∞ − ∞, which we cannot use. So we rationalize the numerator (the denominator is 1) to get an expression that we can use the limit laws on. (√ ) (√ ) √4x2 + 17 + 2x lim 4x 2 + 17 − 2x = lim 4x 2 + 17 − 2x · √ x→∞ x→∞ 4x2 + 17 + 2x (4x2 + 17) − 4x2 = lim √ x→∞ 4x2 + 17 + 2x 17 = lim √ =0 x→∞ 4x2 + 17 + 2x . . . . . . V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 36 / 38
  81. 81. Kick it up a notch Example (√ ) Compute lim 4x2 + 17x − 2x . x→∞ . . . . . . V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 37 / 38
  82. 82. Kick it up a notch Example (√ ) Compute lim 4x2 + 17x − 2x . x→∞ Solution Same trick, different answer: (√ ) lim 4x2 + 17x − 2x x→∞ (√ √ ) 4x2 + 17 + 2x = lim + 17x − 2x · √ 4x2 x→∞ 4x2 + 17x + 2x (4x2 + 17x) − 4x2 = lim √ x→∞ 4x2 + 17x + 2x 17x 17 17 = lim √ = lim √ = x→∞ 4x2 + 17x + 2x x→∞ 4 + 17/x + 2 4 . . . . . . V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 37 / 38
  83. 83. Summary Infinity is a more complicated concept than a single number. There are rules of thumb, but there are also exceptions. Take a two-pronged approach to limits involving infinity: Look at the expression to guess the limit. Use limit rules and algebra to verify it. . . . . . . V63.0121.021, Calculus I (NYU) Section 1.6 Limits involving Infinity September 23, 2010 38 / 38

×