Lesson 4: Limits Involving Infinity
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Lesson 4: Limits Involving Infinity

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We examine two ways of extending the definition of limit: A function can be said to have a limit of infinity (or minus infinity) at a point if it grows without bound near that point. ...

We examine two ways of extending the definition of limit: A function can be said to have a limit of infinity (or minus infinity) at a point if it grows without bound near that point.
A function can have a limit at a point if values of the function get close to a value as the points get arbitrarily large.

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    Lesson 4: Limits Involving Infinity Lesson 4: Limits Involving Infinity Presentation Transcript

    • Section 2.5 Limits Involving Infinity Math 1a February 4, 2008 Announcements Syllabus available on course website All HW on website now No class Monday 2/18 ALEKS due Wednesday 2/20
    • Outline Infinite Limits Vertical Asymptotes Infinite Limits we Know Limit “Laws” with Infinite Limits Indeterminate Limits Limits at Infinity Algebraic rates of growth Exponential rates of growth Rationalizing to get a limit Worksheet
    • Infinite Limits Definition The notation lim f (x) = ∞ x→a means that the 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. Definition The notation lim f (x) = −∞ x→a means that the values of f (x) can be made arbitrarily large negative by taking x sufficiently close to a but not equal to a. Of course we have definitions for left- and right-hand infinite limits.
    • 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−
    • Infinite Limits we Know 1 lim =∞ x→0+ x 1 lim = −∞ x→0− x 1 lim 2 = ∞ x→0 x
    • Finding limits at trouble spots Example Let t2 + 2 f (t) = t 2 − 3t + 2 Find lim f (t) and lim+ f (t) for each a at which f is not t→a− t→a continuous.
    • Finding limits at trouble spots Example Let t2 + 2 f (t) = t 2 − 3t + 2 Find lim f (t) and lim+ f (t) for each a at which f is not t→a− t→a continuous. Solution The denominator factors as (t − 1)(t − 2). We can record the signs of the factors on the number line.
    • − 0 + (t − 1) 1
    • − 0 + (t − 1) 1 − 0 + (t − 2) 2
    • − 0 + (t − 1) 1 − 0 + (t − 2) 2 + (t 2 + 2)
    • − 0 + (t − 1) 1 − 0 + (t − 2) 2 + (t 2 + 2) f (t) 1 2
    • − 0 + (t − 1) 1 − 0 + (t − 2) 2 + (t 2 + 2) + f (t) 1 2
    • − 0 + (t − 1) 1 − 0 + (t − 2) 2 + (t 2 + 2) + ±∞ f (t) 1 2
    • − 0 + (t − 1) 1 − 0 + (t − 2) 2 + (t 2 + 2) + ±∞ − f (t) 1 2
    • − 0 + (t − 1) 1 − 0 + (t − 2) 2 + (t 2 + 2) + ±∞ − ∞ f (t) 1 2
    • − 0 + (t − 1) 1 − 0 + (t − 2) 2 + (t 2 + 2) + ±∞ − ∞ + f (t) 1 2
    • Limit Laws with infinite limits To aid your intuition The sum of positive infinite limits is ∞. That is ∞+∞=∞ The sum of negative infinite limits is −∞. −∞ − ∞ = −∞ The sum of a finite limit and an infinite limit is infinite. a+∞=∞ a − ∞ = −∞
    • Rules of Thumb with infinite limits Don’t try this at home! The sum of positive infinite limits is ∞. That is ∞+∞=∞ The sum of negative infinite limits is −∞. −∞ − ∞ = −∞ The sum of a finite limit and an infinite limit is infinite. a+∞=∞ a − ∞ = −∞
    • Rules of Thumb with infinite limits The product of a finite limit and an infinite limit is infinite if the finite limit is not 0. ∞ if a > 0 a·∞= −∞ if a < 0. −∞ if a > 0 a · (−∞) = ∞ if a < 0. The product of two infinite limits is infinite. ∞·∞=∞ ∞ · (−∞) = −∞ (−∞) · (−∞) = ∞ The quotient of a finite limit by an infinite limit is zero: a = 0. ∞
    • Indeterminate Limits Limits of the form 0 · ∞ and ∞ − ∞ are indeterminate. There is no rule for evaluating such a form; the limit must be examined more closely.
    • Indeterminate Limits Limits of the form 0 · ∞ and ∞ − ∞ are indeterminate. There is no rule for evaluating such a form; the limit must be examined more closely. 1 Limits of the form are also indeterminate. 0
    • Outline Infinite Limits Vertical Asymptotes Infinite Limits we Know Limit “Laws” with Infinite Limits Indeterminate Limits Limits at Infinity Algebraic rates of growth Exponential rates of growth Rationalizing to get a limit Worksheet
    • 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 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→−∞
    • 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!
    • Theorem Let n be a positive integer. Then 1 lim n = 0 x→∞ x 1 lim =0 x→−∞ x n
    • Using the limit laws to compute limits at ∞ Example Find 2x 3 + 3x + 1 lim x→∞ 4x 3 + 5x 2 + 7 if it exists. A does not exist B 1/2 C 0 D ∞
    • Using the limit laws to compute limits at ∞ Example Find 2x 3 + 3x + 1 lim x→∞ 4x 3 + 5x 2 + 7 if it exists. A does not exist B 1/2 C 0 D ∞
    • Solution Factor out the largest power of x from the numerator and denominator. We have 2x 3 + 3x + 1 x 3 (2 + 3/x 2 + 1/x 3 ) = 3 4x 3 + 5x 2 + 7 x (4 + 5/x + 7/x 3 ) 2x 3 + 3x + 1 2 + 3/x 2 + 1/x 3 lim = lim x→∞ 4x 3 + 5x 2 + 7 x→∞ 4 + 5/x + 7/x 3 2+0+0 1 = = 4+0+0 2
    • Solution Factor out the largest power of x from the numerator and denominator. We have 2x 3 + 3x + 1 x 3 (2 + 3/x 2 + 1/x 3 ) = 3 4x 3 + 5x 2 + 7 x (4 + 5/x + 7/x 3 ) 2x 3 + 3x + 1 2 + 3/x 2 + 1/x 3 lim = lim x→∞ 4x 3 + 5x 2 + 7 x→∞ 4 + 5/x + 7/x 3 2+0+0 1 = = 4+0+0 2 Upshot When finding limits of algebraic expressions at infinitely, look at the highest degree terms.
    • Another Example Example Find √ 3x 4 + 7 lim x→∞ x2 + 3
    • Another Example Example Find √ 3x 4 + 7 lim x→∞ x2 + 3 Solution √ The limit is 3.
    • Example x2 Make a conjecture about lim . x→∞ 2x
    • Example x2 Make a conjecture about lim . x→∞ 2x Solution The limit is zero. Exponential growth is infinitely faster than geometric growth
    • Rationalizing to get a limit Example Compute lim 4x 2 + 17 − 2x . x→∞
    • Rationalizing to get a limit Example Compute lim 4x 2 + 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.
    • Outline Infinite Limits Vertical Asymptotes Infinite Limits we Know Limit “Laws” with Infinite Limits Indeterminate Limits Limits at Infinity Algebraic rates of growth Exponential rates of growth Rationalizing to get a limit Worksheet
    • Worksheet