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Chapter 1: Problem 50

Use a graphing utility to graph the function. Use the graph to determine any \(x\) -values at which the function is not continuous. $$ f(x)=\left\\{\begin{array}{ll} \frac{\cos x-1}{x}, & x<0 \\ 5 x, & x \geq 0 \end{array}\right. $$

Short Answer

Expert verified
The function \(f(x)\) is discontinuous at \(x = 0\), since the left limit \(\lim_{{x \to 0^-}} \frac{\cos x - 1}{x}\) does not exist, and the right limit \(\lim_{{x \to 0^+}} 5x\) is \(0\), but aren't equal.

Step by step solution

01

- Graph the Functions

Use the graphing utility to first graph the function \(\frac{\cos x-1}{x}\) and next graph the function \(5x\). Observe where these two graphs meet or where they do not meet.
02

- Check for Discontinuity for \(x < 0\)

To find the \(x\) values where function is not continuous for \(x < 0\), check the limits: \(\lim_{{x \to 0^-}} \frac{\cos x - 1}{x}\). If the limit is not defined or not equal to the function value at \(x = 0\), then function is discontinuous.
03

- Check for Discontinuity for \(x \geq 0\)

To determine any \(x\) values that result in the function being not continuous for \(x \geq 0\), use the limit approach: \(\lim_{{x \to 0^+}} 5x\). If this limit does not equal the function value at \(x = 0\), then the function can be considered discontinuous.

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