Ma 113 Functions And Inverse Functions, The Exponential Function And The Logarithm Worksheet Page 26
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41Worksheet # 20: Limits at Infinity & L’Hˆ o pital’s Rule
1. (a) Describe the behavior of the function f (x) if lim f (x) = L and lim f (x) = M .
(b) Explain the difference between “ lim
f (x) =
” and “ lim f (x) =
3”.
3
2. Evaluate the following limits, or explain why the limit does not exist:
2
3x
7x
(d)
lim 3
(a) lim
x
8
3
2
5x
7x
+ 9
2
2x
6
(e)
lim
(b) lim
2
3
x
8x
8999
4
x
8x + 9
10
x
x
+ 2x
(c)
lim
(f)
lim
6
2
5
x
4x
x
2
2
x
x
3. Find the limits lim f (x) and lim f (x) if f (x) =
.
x + 1
x
1
4. Sketch a graph with all of the following properties:
lim f (t) = 2
lim
f (t) =
0
lim f (t) = 0
lim
f (t) = 3
4
lim
f (t) =
f (4) = 6
0
5. Find the following limits;
2
3x + 2 x
5x
+ sin x
(a) lim
(c) lim
2
1
x
3x
+ cos x
2x
5
(b)
lim
3x + 2
6. Carefully state l’Hˆ o pital’s Rule.
7. Compute the following limits. Use l’Hˆ o pital’s Rule where appropriate but first check that no easier
method will solve the problem.
9
2
x
1
x
+ x
6
(a) lim
(c) lim
5
x
1
x
2
1
2
sin(4x)
2
x
+ 2x
2
(b) lim
(d) lim
tan(5x)
2
0
x
2x + 2
1
8. Find the value A for which we can use l’Hˆ o pital’s rule to evaluate the limit
2
x
+ Ax
2
lim
.
x
2
2
For this value of A, give the value of the limit.
9. Compute the following limits. Use l’Hˆ o pital’s Rule where appropriate but first check that no easier
method will solve the problem.
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