212
Chapter 2 | Limits
228. lim x →1
(8 x +16) =24
3 x 2 −2 x +4
214. lim x →0
x 3 =0
229. lim x →0
x 3 −2 x 2 −1 3 x −2
215. lim x →3
230. A ball is thrown into the air and the vertical position is givenby x ( t ) =−4.9 t 2 +25 t +5. Use the Intermediate Value Theorem to show that the ball must land on the ground sometime between 5 sec and 6 sec after the throw. 231. A particle moving along a line has a displacement according to the function x ( t ) = t 2 −2 t +4, where x is measured in meters and t is measured in seconds. Find the average velocity over the time period t = [0, 2]. 232. From the previous exercises, estimate the instantaneous velocity at t =2 by checking the average velocity within t =0.01sec.
cot x cos x
lim x → π /2
216.
x 2 +25 x +5
lim x →−5
217.
3 x 2 −2 x −8 x 2 −4
218. lim x →2
x 2 −1 x 3 −1
219. lim x →1
x 2 −1 x −1
220. lim x →1
4− x
221. lim x →4
x −2
1 x −2
222. lim x →4
In the following exercises, use the squeeze theorem to prove the limit. 223. lim x →0 x 2 cos(2 πx ) =0
x 3 sin ⎛
⎞ ⎠ =0
⎝ π x
224. lim x →0
225. Determine the domain such that the function f ( x ) = x −2+ xe x is continuous over its domain.
In the following exercises, determine the value of c such that the function remains continuous. Draw your resulting function to ensure it is continuous. 226. f ( x ) = ⎧ ⎩ ⎨ x 2 +1, x > c 2 x , x ≤ c
⎧ ⎩
⎨ x +1, x >−1 x 2 + c , x ≤ −1
227. f ( x ) =
In the following exercises, use the precise definition of limit to prove the limit.
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