Question

In Exercises \(7-20,\) find the vertical asymptotes (if any) of the function. $$ f(x)=\frac{4}{(x-2)^{3}} $$

Short answer

The vertical asymptote of the function \(f(x)=\frac{4}{(x-2)^{3}}\) is \(x = 2\).

Step-by-step solution

Identify the Denominator of the Function

The denominator of the function \(f(x)=\frac{4}{(x-2)^{3}}\) is \((x-2)^{3}\). This will be used to find values of x that make the function undefined.

Determine the Value that Makes the Denominator Zero

Set the denominator \((x-2)^{3}\) equal to zero and solve for x. Therefore, \(x - 2 = 0\) when simplified, results in \(x = 2\).

Identify the Vertical Asymptote

The x-value that makes the denominator zero is the vertical asymptote of the function. Here, the vertical asymptote is \(x = 2\).