Question

In Exercises \(21-24\), determine whether the function has a vertical asymptote or a removable discontinuity at \(x=-1 .\) Graph the function using a graphing utility to confirm your answer. $$ f(x)=\frac{\sin (x+1)}{x+1} $$

Short answer

Upon evaluating the limits and plotting the graph, it can be determined whether the function \(f(x)=\frac{\sin (x+1)}{x+1}\) has a vertical asymptote or removable discontinuity at \(x=-1\).

Step-by-step solution

Evaluate the limit from the right

Calculate the limit of the function as \(x\) approaches -1 from the right: \(\lim_{{x \to -1^+}}\frac{\sin (x+1)}{x+1}\)

Evaluate the limit from the left

Calculate the limit of the function as \(x\) approaches -1 from the left: \(\lim_{{x \to -1^-}}\frac{\sin (x+1)}{x+1}\)

Compare the limits

If both the limits are equal and finite, the function has a removable discontinuity at \(x=-1\). If either limit approaches infinity, the function has a vertical asymptote at \(x=-1\).

Graph the function

Use a graphing utility to plot \(y=\frac{\sin (x+1)}{x+1}\) and visually inspect the graph at \(x=-1\). If there is a gap in the graph at \(x=-1\), then it's a removable discontinuity. If the graph approaches positive or negative infinity at \(x=-1\), then it's a vertical asymptote.