470
Chapter 4 | Applications of Derivatives
4.8 EXERCISES For the following exercises, evaluate the limit. 356. Evaluate the limit lim x →∞ e x x . 357. Evaluate the limit lim x →∞ .
x −1 sin x
372. lim x →1
(1+ x ) n −1 x
373. lim x →0
e x x k
(1+ x ) n −1− nx x 2
374. lim x →0
ln x x k
358. Evaluate the limit lim x →∞
.
sin x −tan x x 3
375. lim x →0
x − a x 2 − a 2 x − a x 3 − a 3 x − a x n − a n
359. Evaluate the limit lim x → a
, a ≠0 .
1+ x − 1− x x
376. lim x →0
360. Evaluate the limit lim x → a
, a ≠0 .
e x − x −1 x 2
377. lim x →0
361. Evaluate the limit lim x → a
, a ≠0 .
tan x x
378. lim x →0
For the following exercises, determine whether you can apply L’Hôpital’s rule directly. Explain why or why not. Then, indicate if there is some way you can alter the limit so you can apply L’Hôpital’s rule.
x −1 ln x
379. lim x →1
x 2 ln x
( x +1) 1/ x
lim x →0 +
362.
380. lim x →0
x − x 3 x −1
x 1/ x
lim x
363.
381. lim x →1
→∞
x 2/ x
364. lim x →0
x 2 x
lim x →0 +
382.
x 2 1/ x
⎛ ⎝ 1 x
⎞ ⎠
365. lim x →0
lim x
x sin
383.
→∞
e x x
sin x − x x 2
lim x
366.
384. lim x →0
→∞
For the following exercises, evaluate the limits with either L’Hôpital’s rule or previously learned methods.
⎛ ⎝ x 4
⎞ ⎠
lim x →0 +
x ln
385.
x 2 −9 x −3 x 2 −9 x +3
367. lim x →3
x − e x )
→∞ (
lim x
386.
x 2 e − x
lim x
387.
368. lim x →3
→∞
3 x −2 x x
388. lim x →0
(1+ x ) −2 −1 x
369. lim x →0
1+1/ x 1−1/ x
389. lim x →0
cos x π 2 − x
lim x → π /2
370.
(1−tan x )cot x
lim x → π /4
390.
x − π sin x
371. lim x → π
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