Question

In Exercises \(75-86\), use a computer algebra system to analyze the graph of the function. Label any extrema and/or asymptotes that exist. $$ f(x)=\frac{2 \sin 2 x}{x} $$

Short answer

The function \(f(x)=\frac{2 \sin 2x}{x}\) doesn't have any vertical asymptotes. The x-axis, y = 0, is the horizontal asymptote. The extrema can be found by determining the critical points from the derivative of the function.

Step-by-step solution

Analyze the function and find vertical asymptotes

For rational functions like this one, a vertical asymptote can be found wherever the denominator is zero and the numerator is not simultaneously zero. The denominator, x, equals zero at the point x=0. However, at x = 0, sin(2x) will also be zero, so there is no vertical asymptote.

Find the limit for horizontal asymptotes

We need to analyze the limit of the function when x approaches infinity or minus infinity. After the calculation, the limit results in zero when x approaches both infinity and minus infinity. Therefore, the x-axis (y = 0) is the horizontal asymptote.

Find the extrema

The extrema of the function can be found where its derivative equals to zero or does not exist. Find the derivative of the function and solve for x. Then substitute these values into the original function to get the respective y values. This helps to locate the extrema on the graph of the function.