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Chapter 3: Problem 22

Find the vertical asymptotes, if any, and the values of \(x\) corresponding to holes, if any, of the graph of each rational function. $$f(x)=\frac{x}{x-3}$$

Short Answer

Expert verified
The rational function \(f(x)=\frac{x}{x-3}\) has a vertical asymptote at \(x=3\) and no holes.

Step by step solution

01

Identify the denominator

The function being considered is \(f(x)=\frac{x}{x-3}\). The denominator of this function is \(x-3\).
02

Find the values of \(x\) that make the denominator zero

To find the vertical asymptotes, set the denominator equal to zero and solve for \(x\). \[x-3=0\] Solving this equation, we find that \(x=3\).
03

Determine if there are holes

A hole can appear in a rational function when a factor appears in both the numerator and denominator. But, in this case, the numerator is a constant and it doesn't have any factors that can cancel out with the denominator. So, there are no holes for this function.
04

Identify the vertical asymptote

Since there are no removable discontinuities (holes), the zero in the denominator results in a vertical asymptote. Therefore, the vertical asymptote of this function is \(x=3\).

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