1. BMM 104: ENGINEERING MATHEMATICS I Page 1 of 6
CHAPTER 2: LIMITS
Calculating Limits Using the Limit Laws
THEOREM Limit Laws
If L, M, c and k are real numbers and
lim f ( x ) = L and lim g ( x ) = M , then
x →c x →c
1. Sum Rule: lim( f ( x ) + g ( x )) = L + M
x →c
2. Difference Rule: lim( f ( x ) − g( x )) = L − M
x →c
3. Product Rule: lim( f ( x ) • g( x )) = L • M
x→ c
4. Constant Multiple Rule: lim( k • f ( x )) = k • L
x→ c
f(x) L
5. Quotient Rule: lim = ,M ≠ 0
x →c g( x ) M
6. Power Rule: If r and s are integers with no common factor and s ≠ 0 , then
lim( f ( x )) r / s = Lr / s
x→c
Example: Attend lecture.
PROBLEM SET: CHAPTER 2
Find the following limits.
lim ( 2 x + 5 ) x −5
1. 19. lim
x →−7 x →5 x 2 − 25
lim ( 10 − 3 x ) x+3
2. 20. lim
x →12 x →−3 x + 4 x + 3
2
lim( − x 2 + 5 x − 2 ) x + 3 x − 10
2
3. 21. lim
x→ 2 x →−5 x +5
lim ( x 3 − 2 x 2 + 4 x + 8 ) x − 7 x + 10
2
4. 22. lim
x → −2 x →2 x −2
lim 8( t − 5 )( t − 7 ) t2 +t −2
5. t →6
23. lim 2
t →1 t −1
lim 3 s( 2 s − 1 ) t 2 + 3t + 2
6. s →2 / 3
24. lim
t → −1 t 2 − t − 2
x +3 − 2x − 4
7. lim 25. lim
x →2 x +6 x →−2 x 3 + 2 x 2
2. BMM 104: ENGINEERING MATHEMATICS I Page 2 of 6
4 5y3 +8 y2
8. lim 26. lim
x→5 x −7 y →0 3 y 4 − 16 y 2
lim
y2 u4 −1
9. y→ 5 5 − y
27. lim 3
− u →1 u − 1
y +2 v3 − 8
10. lim 28. lim 4
y →2 y +5y +6
2
v →2 v − 16
lim 3( 2 x − 1 ) 2 x −3
11. x →−1
29. lim
x→9 x −9
lim ( x + 3 )1984 4x − x2
12. 30. lim
x →−4 x →4 2− x
x −1
13. lim ( 5 − y ) 4/3
31. lim
y →−3 x →1 x +3 −2
lim( 2 z − 8 )1 / 3 x2 + 8 − 3
14. z →0
32. lim
x →−1 x +1
3 x + 12 − 4
2
15. lim 33. lim
h →03h + 1 + 1 x →2 x −2
5 x +2
16. lim 34. lim
h →0 5h + 4 + 2 x →−2
x2 + 5 −3
3h + 1 − 1 2 − x2 −5
17. lim 35. lim
h →0 h x →−3 x +3
5h + 4 − 2 4−x
18. lim 36. lim
h →0 h x →4
5 − x2 + 9
ANSWERS FOR PROBLEM SET: CHAPTER 2
Find the following limits.
1. -9 19. 1/10
2. - 26 20. - 1/2
3. 4 21. -7
4. - 16 22. -3
5. -8 23. 3/2
6. 2/3 24. - 1/3
7. 5/8 25. - 1/2
8. -2 26. - 1/2
9. 5/2 27. 4/3
10. 1/5 28. 3/8
11. 27 29. 1/6
12. 1 30. 16
13. 16 31. 4
14. -2 32. - 1/3
15. 3/2 33. 1/2
3. BMM 104: ENGINEERING MATHEMATICS I Page 3 of 6
16. 5/4 34. - 3/2
17. 3/2 35. 3/2
18. 5/4 36. 5/4
One-Sided Limits and Limits at Infinity
THEOREM
A function f(x) has a limit as x approaches c if and only if it has left-hand and right-hand
limits there and these one-sided limits are equal:
lim f ( x ) = L ⇔ lim f ( x ) = L and lim f ( x ) = L .
x →c x →c − x →c +
THEOREM Limit Laws as x → ±∞
If L, M, and k are real numbers and
lim f ( x ) = L and lim g ( x ) = M , then
x →±∞ x →±∞
1. Sum Rule: lim ( f ( x ) + g( x )) = L + M
x → ±∞
2. Difference Rule: lim ( f ( x ) − g( x )) = L − M
x →±∞
3. Product Rule: lim ( f ( x ) • g( x )) = L • M
x→ ± c
4. Constant Multiple Rule: lim ( k • f ( x )) = k • L
x → ±∞
f(x) L
5. Quotient Rule: lim = ,M ≠ 0
x →±∞ g( x ) M
6. Power Rule: If r and s are integers with no common factor and s ≠ 0 , then
lim ( f ( x )) r / s = Lr / s
x→± c
Example: Attend lecture.
PROBLEM SET: CHAPTER 2
Finding One-Sided Limits Algebraically
4. BMM 104: ENGINEERING MATHEMATICS I Page 4 of 6
Find the following limits.
x +2
1. lim −
x →−0.5 x +1
x −1
2. lim
x →1+ x+2
x 2 x + 5
3. lim 2
x → −2 + x + 1 x + x
1 x + 6 3 − x
4. lim
x →1− x + 1 x 7
h 2 + 4h + 5 − 5
5. lim+
h →0 h
6 − 5 h 2 + 11h + 6
6. lim−
h →0 h
x +2 x +2
7. a. lim ( x + 3) b. lim− ( x + 3 )
x →−2 +
x +2 x →−2 x +2
2 x ( x − 1) 2 x ( x − 1)
8. a. lim b. lim
x→ +
1 x −1 x→ −
1 x −1
sin θ
lim
Using θ →0 =1
θ
Find the following limits.
sin 2θ x + x cos x
1. lim 9. lim
θ→0 2θ x →0 sin x cos x
sin kt x 2 − x + sin x
2. lim 10. lim
t→0 t x →0 2x
sin 3 y sin( 1 − cos t )
3. lim 11. lim
y→0 4y t →0 1 − cos t
h sin(sinh)
4. lim− 12. lim
h →0 sin 3h h→ 0 sinh
tan 2 x sin θ
5. lim 13. lim
x→0 x θ →0 sin 2θ
2t sin 5 x
6. lim 14. lim
t → tan t
0 x→ sin 4 x
0
x csc 2 x tan 3 x
7. lim 15. lim
x→0 cos 5 x x→ sin 8 x
0
lim 6 x 2 ( cot x )( csc 2 x )
sin 3 y cot 5 y
8. 16. lim
x→0 y→0 y cot 4 y
5. BMM 104: ENGINEERING MATHEMATICS I Page 5 of 6
Limits of Rational Functions
In the following questions, find the limit of each rational function (a) as x → ∞ and (b)
as x → −∞ .
2x + 3 1
1. f(x)= 6. g( x ) =
5x +7 x − 4x + 1
3
2x3 + 7 10 x + x 4 + 31
5
2. f(x)= 7. g( x ) =
x3 − x2 + x + 7 x6
x+1 9x4 + x
3. f(x)= 2 8. h( x ) = 4
x +3 2x + 5x2 − x + 6
3x + 7 − 2x3 − 2x + 3
4. f(x)= 2 9. h( x ) = 3
x −2 3x + 3x 2 − 5 x
7 x3 − x4
5. h( x ) = 3 10. h( x ) = 4
x − 3x 2 + 6 x x − 7 x3 + 7 x2 + 9
Limits with Noninteger or Negative Powers
Find the following limits.
2 x + x −1 x −1 + x −4
1. lim 4. lim
x →∞ 3 x −7 x →∞ x −2 − x −3
2+ x 2x5 / 3 − x1 / 3 + 7
2. lim 5. lim
x→∞ 2− x x →∞ x 8 / 5 + 3x + x
3
x −5 x 3
x − 5x + 3
3. lim 6. lim
x →−∞ 3 x+ x
5 x →∞ 2x + x 2 / 3 − 4
ANSWERS FOR PROBLEM SET: CHAPTER 2
Finding One-Sided Limits Algebraically
1. 3 4. 1
2
2. 0 5.
5
