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Chapter 2 | Limits
154. Consider the graph of the function y = f ( x ) shown in the following graph.
158. Sketch the graph of the function y = f ( x ) with properties i. through iv. i. The domain of f is ⎡ ⎣ 0, 5 ⎤ ⎦ . ii. lim x →1 + f ( x ) and lim x →1 − f ( x ) exist and are equal. iii. f ( x ) is left continuous but not continuous at x =2, and right continuous but not continuous at x =3. iv. f ( x ) has a removable discontinuity at x =1, a jump discontinuity at x =2, and the following limits hold: lim x →3 − f ( x ) =−∞ and lim x →3 + f ( x ) =2. In the following exercises, suppose y = f ( x ) is defined for all x . For each description, sketch a graph with the indicated property. 159. Discontinuous at x =1 with lim x →−1 f ( x ) =−1 and lim x →2 f ( x ) =4 160. Discontinuous at x =2 but continuous elsewhere with lim x →0 f ( x ) = 1 2 Determine whether each of the given statements is true. Justify your response with an explanation or counterexample. 161. f ( t ) = 2 e t − e − t is continuous everywhere. 162. If the left- and right-hand limits of f ( x ) as x → a exist and are equal, then f cannot be discontinuous at x = a . 163. If a function is not continuous at a point, then it is not defined at that point. 164. According to the IVT, cos x −sin x − x =2 has a solution over the interval [−1, 1]. 165. If f ( x ) is continuous such that f ( a ) and f ( b ) have opposite signs, then f ( x ) =0 has exactly one solution in ⎡ ⎣ a , b ⎤ ⎦ . 166. The function f ( x ) = x 2 −4 x +3 x 2 −1 is continuous over the interval [0, 3].
a. Find all values for which the function is discontinuous. b. For each value in part a., state why the formal definition of continuity does not apply. c. Classify each discontinuity as either jump, removable, or infinite.
⎧ ⎩ ⎨ 3 x , x >1 x 3 , x <1 . a. Sketch the graph of f .
155. Let f ( x ) =
b. Is it possible to find a value k such that f (1) = k , which makes f ( x ) continuous for all real numbers? Briefly explain. 156. Let f ( x ) = x 4 −1 x 2 −1 for x ≠ −1, 1. a. Sketch the graph of f . b. Is it possible to find values k 1 and k 2 such that f (−1) = k 1 and f (1) = k 2 , and that makes f ( x ) continuous for all real numbers? Briefly explain. 157. Sketch the graph of the function y = f ( x ) with properties i. through vii. i. The domain of f is (−∞, +∞). ii. f has an infinite discontinuity at x =−6. iii. f (−6) =3 iv. lim x →−3 − f ( x ) = lim x →−3 + f ( x ) =2 v. f (−3) =3 vi. f is left continuous but not right continuous at x =3. vii. lim x → −∞ f ( x ) =−∞ and lim x → +∞ f ( x ) =+∞
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