Limits and Rates of Change - The Questions - 1,001 Calculus Practice Problems

1,001 Calculus Practice Problems

Part I

The Questions

Chapter 3

Limits and Rates of Change

Limits are the foundation of calculus. Being able to work with limits and to understand them conceptually is crucial, because key ideas and definitions in calculus make use of limits. This chapter examines a variety of limit problems and makes the intuitive idea of continuity formal by using limits. Many later problems also involve the use of limits, so although limits may go away for a while during your calculus studies, they'll return!

The Problems You'll Work On

In this chapter, you encounter a variety of problems involving limits:

· Using graphs to find limits

· Finding left-hand and right-hand limits

· Determining infinite limits and limits at infinity

· Practicing many algebraic techniques to evaluate limits of the form 0/0

· Determining where a function is continuous

What to Watch Out For

You can use a variety of techniques to evaluate limits, and you want to be familiar with them all! Remember the following tips:

· When substituting in the limiting value, a value of zero in the denominator of a fraction doesn't automatically mean that the limit does not exist! For example, if the function has a removable discontinuity, the limit still exists!

· Be careful with signs, as you may have to include a negative when evaluating limits at infinity involving radicals (especially when the variable approaches negative infinity). It's easy to make a limit positive when it should have been negative!

· Know and understand the definition of continuity, which says the following: A function f(x) is continuous at a if 9781118496718-eq03001a.png.

Finding Limits from Graphs

167–172 Use the graph to find the indicated limit.

167. 9781118496718-un0302.tif

9781118496718-eq03001.eps

168. 9781118496718-un0303.tif

9781118496718-eq03002.eps

169. 9781118496718-un0304.tif

9781118496718-eq03003.eps

170. 9781118496718-un0305.tif

9781118496718-eq03004.eps

171. 9781118496718-un0306.tif

9781118496718-eq03005.eps

172. 9781118496718-un0306.tif

9781118496718-eq03006.eps

Evaluating Limits

173–192 Evaluate the given limit.

173. 9781118496718-eq03007.eps

174. 9781118496718-eq03008.eps

175. 9781118496718-eq03009.eps

176. 9781118496718-eq03010.eps

177. 9781118496718-eq03011.eps

178. 9781118496718-eq03012.eps

179. 9781118496718-eq03013.eps

180. 9781118496718-eq03014.eps

181. 9781118496718-eq03015.eps

182. 9781118496718-eq03016.eps

183. 9781118496718-eq03017.eps

184. 9781118496718-eq03018.eps

185. 9781118496718-eq03019.eps

186. 9781118496718-eq03020.eps

187. 9781118496718-eq03021.eps

188. 9781118496718-eq03022.eps

189. 9781118496718-eq03023.eps

190. 9781118496718-eq03024.eps

191. 9781118496718-eq03025.eps

192. 9781118496718-eq03026.eps

Applying the Squeeze Theorem

193–198 Use the squeeze theorem to evaluate the given limit.

193. If 5 ≤ f (x) ≤ x2 + 3x – 5 for all x, find 9781118496718-eq03027.eps.

194. If x2 + 4 ≤ f (x) ≤ 4 + sin x for –2 ≤ x ≤ 5, find 9781118496718-eq03028.png

195. If 2xf (x) ≤ x3 + 1 for 0 ≤ x ≤ 2, evaluate 9781118496718-eq03029.png

196. Find the limit: 9781118496718-eq03030.png

197. Find the limit: 9781118496718-eq03031.png

198. Find the limit: 9781118496718-eq03032.png

Evaluating Trigonometric Limits

199–206 Evaluate the given trigonometric limit. Recall that 9781118496718-eq03033.png and that 9781118496718-eq03034.png.

199. 9781118496718-eq03035.eps

200. 9781118496718-eq03036.eps

201. 9781118496718-eq03037.eps

202. 9781118496718-eq03038.eps

203. 9781118496718-eq03039.eps

204. 9781118496718-eq03040.eps

205. 9781118496718-eq03041.eps

206. 9781118496718-eq03042.eps

Infinite Limits

207–211 Find the indicated limit using the given graph.

207. 9781118496718-un0308.tif

9781118496718-eq03043.eps

208. 9781118496718-un0309.tif

9781118496718-eq03044.eps

209. 9781118496718-un0310.tif

9781118496718-eq03045.eps

210.

9781118496718-un0311.tif

9781118496718-eq03046.eps

211. 9781118496718-un0311.tif

9781118496718-eq03047.eps

212−231 Find the indicated limit.

212. 9781118496718-eq03048.eps

213. 9781118496718-eq03049.eps

214. 9781118496718-eq03050.eps

215. 9781118496718-eq03052.eps

216. 9781118496718-eq03053.eps

217. 9781118496718-eq03054.eps

218. 9781118496718-eq03055.eps

219. 9781118496718-eq03056.eps

220. 9781118496718-eq03057.eps

221. 9781118496718-eq03058.eps

222. 9781118496718-eq03059.eps

223. 9781118496718-eq03061.eps

224. 9781118496718-eq03062.eps

225. 9781118496718-eq03064.eps

226. 9781118496718-eq03065.eps

227. 9781118496718-eq03066.eps

228. 9781118496718-eq03067.eps

229. 9781118496718-eq03068.eps

230. 9781118496718-eq03069.eps

231. 9781118496718-eq03070.eps

Limits from Graphs

232–235 Find the indicated limit using the given graph.

232. 9781118496718-un0313.tif

9781118496718-eq03071.eps

233.

9781118496718-un0314.tif

9781118496718-eq03072.eps

234. 9781118496718-un0315.tif

9781118496718-eq03073.eps

235. 9781118496718-un0315.tif

9781118496718-eq03074.eps

Limits at Infinity

236–247 Find the indicated limit.

236. 9781118496718-eq03075.eps

237. 9781118496718-eq03076.eps

238. 9781118496718-eq03077.eps

239. 9781118496718-eq03078.eps

240. 9781118496718-eq03079.eps

241. 9781118496718-eq03080.eps

242. 9781118496718-eq03081.eps

243. 9781118496718-eq03082.eps

244. 9781118496718-eq03083.eps

245. 9781118496718-eq03084.eps

246. 9781118496718-eq03085.eps

247. 9781118496718-eq03086.eps

Horizontal Asymptotes

248–251 Find any horizontal asymptotes of the given function.

248. 9781118496718-eq03087.eps

249. 9781118496718-eq03088.eps

250. 9781118496718-eq03089.eps

251. 9781118496718-eq03090.eps

Classifying Discontinuities

252–255 Use the graph to find all discontinuities and classify each one as a jump discontinuity, a removable discontinuity, or an infinite discontinuity.

252. 9781118496718-un0316.tif

253. 9781118496718-un0317.tif

254. 9781118496718-un0318.tif

255. 9781118496718-un0319.tif

Continuity and Discontinuities

256–261 Determine whether the function is continuous at the given value of a. If it's continuous, state the value at f (a). If it isn't continuous, classify the discontinuity as a jump, removable, or infinite discontinuity.

256. 9781118496718-eq03091.eps

where a = 2

257. 9781118496718-eq03092.eps

where a = 1

258. 9781118496718-eq03093.eps

where a = 3

259. 9781118496718-eq03094.eps

where a = 16

260. 9781118496718-eq03095.eps

where a = –6

261. 9781118496718-eq03096.eps

where a = –1

262–265 Determine whether the function is continuous at the given values of a. If it isn’t continuous, classify each discontinuity as a jump, removable, or infinite discontinuity.

262. 9781118496718-eq03097.eps

where a = 0 and a = π

263. 9781118496718-eq03098.eps

where a = 1 and a = 3

264. 9781118496718-eq03099.eps

where a = 2 and a = 3

265. 9781118496718-eq03100.eps

where a = 0 and a = 4

Making a Function Continuous

266–267 Determine the value of c that makes the given function continuous everywhere.

266. 9781118496718-eq03101.eps

267. 9781118496718-eq03102.eps

The Intermediate Value Theorem

268–271 Determine which of the given intervals is guaranteed to contain a root of the function by the intermediate value theorem.

268. By checking only the endpoints of each interval, determine which interval contains a root of the function 9781118496718-eq03103.eps by the intermediate value theorem:

(A) [–5, –4]

(B) [–4, –3]

(C) [0, 1]

(D) [1, 2]

(E) [5, 12]

269. By checking only the endpoints of each interval, determine which interval contains a root of the function 9781118496718-eq03104.eps by the intermediate value theorem:

(A) [0, 1]

(B) [1, 4]

(C) [4, 9]

(D) [9, 16]

(E) [16, 25]

270. By checking only the endpoints of each interval, determine which interval contains a solution to the equation 2(3x) + x2 – 4 = 32 according to the intermediate value theorem:

(A) [0, 1]

(B) [1, 2]

(C) [2, 3]

(D) [3, 4]

(E) [4, 5]

271. By checking only the endpoints of each interval, determine which interval contains a solution to the equation 9781118496718-eq03105.eps according to the intermediate value theorem:

(A) [0, 1]

(B) [1, 2]

(C) [2, 3]

(D) [3, 4]

(E) [4, 5]