Calculus- Early Transcendentals, 2021a

86 Limits
Exercises for 3.4
Exercise 3.4.1 Compute the limits. If a limit does not exist, explain why.
x 2 + x − 12
(a) lim
x→3 x − 3
x 2 + x − 12
(b) lim
x→1 x − 3
(c)
x 2 + x − 12
lim
x→−4 x − 3
x 2 + x − 12
(d) lim
x→2 x − 2
√ x + 8 − 3
(e) lim
x→1 x − 1
√ √
1 1
(f) lim
x→0 + x + 2 − x
(g) lim 3
x→2
(h) lim
x→4
3x 3 − 5x
(i) lim
x→0
4x − 5x 2
x − 1
(j) lim
x→1
x 2 − 1
x − 1
(k)
√
2 − x 2
lim
x→0 + x
√
2 − x 2
(l) lim
x→0 + x + 1
x 3 − a 3
(m) lim
x→a x − a
(n) lim
x→2
(x 2 + 4) 3
{
1 if x ≠ 0
Exercise 3.4.2 Let f (x) =
0 if x = 0 and g(x) =0. What are the values of L = lim x→0 g(x) and
M = lim x→L f (x)? Is it true that lim x→0 f (g(x)) = M? What are some noteworthy differences between this
example and Theorem 3.11?
3.5 Infinite Limits and Limits at Infinity
We occasionally want to know what happens to some quantity when a variable gets very large or “goes to
infinity”.
Example 3.15: Limit at Infinity
What happens to the function cos(1/x) as x goes to infinity? It seems clear that as x gets larger
and larger, 1/x gets closer and closer to zero, so cos(1/x) should be getting closer and closer to
cos(0)=1.
lim
x→∞
As with ordinary limits, this concept of “limit at infinity” can be made precise. Roughly, we want
f (x)=L to mean that we can make f (x) as close as we want to L by making x large enough.