Page 158 - Calculus Demystified

145
Indeterminate Forms
CHAPTER 5
Exercises
1. If possible, use l’Hôpital’s Rule to evaluate each of the following limits. In
each case, check carefully that the hypotheses of l’Hôpital’s Rule apply.
cos x − 1
(a) lim
2
x→0 x − x 3
e 2x − 1 − 2x
(b) lim
2
x→0 x + x 4
cos x
(c) lim
x→0 x 2
2
[ln x]
(d) lim
x→1 (x − 1)
(x − 2) 3
(e) lim
x→2 sin(x − 2) − (x − 2)
x
e − 1
(f) lim
x→1 x − 1
2. If possible, use l’Hôpital’s Rule to evaluate each of the following limits. In
each case, check carefully that the hypotheses of l’Hôpital’s Rule apply.

x 3
(a) lim
x
x→+∞ e − x 2
ln x
(b) lim
x→+∞ x
e −x
(c) lim
x→+∞ ln[x/(x + 1)]
sin x
(d) lim
x→+∞ e −x
e x
(e) lim
x→−∞ 1/x
ln |x|
(f) lim
x→−∞ e −x
3. If possible, use some algebraic manipulations, plus l’Hôpital’s Rule, to
evaluate each of the following limits. In each case, check carefully that the
hypotheses of l’Hôpital’s Rule apply.
3 −x
(a) lim x e
x→+∞
(b) lim x · sin[1/x]
x→+∞