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Chapter 1: Problem 48

Use a graphing utility to graph the function. Use the graph to determine any \(x\) -values at which the function is not continuous. $$ h(x)=\frac{1}{x^{2}-x-2} $$

Short Answer

Expert verified
The function \(h(x) = \frac{1}{x^{2}-x-2}\) is not continuous at \(x = 2\) and \(x = -1\).

Step by step solution

01

Identify the Critical Points

Set the denominator of the function equal to zero and solve for \(x\). For this function, \(x^{2}-x-2 = 0\). Use the quadratic formula to solve for \(x\), which gives \(x = 2\) and \(x = -1\).
02

Graph the Function

Use a graphing utility to graph the function \(h(x)=\frac{1}{x^{2}-x-2}\). It can be observed that there will be vertical asymptotes at the critical points, \(x = 2\) and \(x = -1\), since the value of \(h(x)\) approaches infinity as \(x\) approaches these values.
03

Identify the Points of Discontinuity

From the graph, it can be observed that vertical asymptotes occur at \(x = 2\) and \(x = -1\). Thus, these are the x-values at which the function is not continuous.

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