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Chapter 2 | Limits
2.5 EXERCISES In the following exercises, write the appropriate ε − δ definition for each of the given statements. 176. lim x → a f ( x ) = N 177. lim t → b g ( t ) = M 178. lim x → c h ( x ) = L 179. lim x → a φ ( x ) = A The following graph of the function f satisfies lim x →2 f ( x ) =2. In the following exercises, determine a value of δ >0 that satisfies each statement.
182. If 0< | x −3| < δ , then | f ( x )+1 | <1. 183. If 0< | x −3| < δ , then | f ( x )+1 | <2. The following graph of the function f satisfies lim x →3 f ( x ) =2. In the following exercises, for each value of ε , find a value of δ >0 such that the precise definition of limit holds true.
180. If 0< | x −2| < δ , then | f ( x )−2 | <1. 181. If 0< | x −2| < δ , then | f ( x )−2 | <0.5. The following graph of the function f satisfies lim x →3 f ( x ) =−1. In the following exercises, determine a value of δ >0 that satisfies each statement.
184. ε =1.5 185. ε =3 [T] In the following exercises, use a graphing calculator to find a number δ such that the statements hold true. 186. | sin(2 x )− 1 2 | <0.1, whenever | x − π 12 | < δ
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