Chapter 2: Problem 4
In Exercises \(1-10,\) find the points of continuity and the points of discontinuity of the function. Identify each type of discontinuity. $$y=|x-1|$$
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In Exercises \(15 - 18\) , explain why you cannot use substitution to determine the limit. Find the limit if it exists. $$\lim _ { x \rightarrow 0 } \frac { ( 4 + x ) ^ { 2 } - 16 } { x }$$
Multiple Choice Which of the following statements about the function \(f(x)=\left\\{\begin{array}{ll}{2 x,} & {0 < x < 1} \\ {1,} & {x=1} \\\ {-x+3,} & {1 < x < 2}\end{array}\right.\) is not true? (A) \(f(1)\) does not exist. (B) \(\lim _{x \rightarrow 0^{+}} f(x)\) exists. (C) \(\lim _{x \rightarrow 2^{-}} f(x)\) exists. (D) \(\lim _{x \rightarrow 1} f(x)\) exists. (E) \(\lim _{x \rightarrow 1} f(x)=f(1)\)
In Exercises \(7 - 14 ,\) determine the limit by substitution. Support graphically. $$\lim _ { x \rightarrow - 1 / 2 } 3 x ^ { 2 } ( 2 x - 1 )$$
Explorations In Exercises 4 and \(42,\) complete the following for the function. (a) Compute the difference quotient \(\frac{f(1+h)-f(1)}{h}\) (b) Use graphs and tables to estimate the limit of the difference quotient in part (a) as \(h \rightarrow 0\) . (c) Compare your estimate in part (b) with the given number. (d) Writing to Learn Based on your computations, do you think the graph of \(f\) has a tangent at \(x=1 ?\) If so, estimate its slope. If not, explain why not. \(f(x)=e^{x}, \quad e\)
Multiple Choice Find the average rate of change of \(f(x)=x^{2}+x\) over the interval \([1,3] .\) . \(\begin{array}{ll}{\text { (A) } y=-2 x} & {\text { (B) } y=2 x \text { (C) } y=-2 x+4} \\ {\text { (D) } y=-x+3} & {\text { (E) } y=x+3}\end{array}\)
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