Question

In Exercises \(37-40\), use a graphing utility to graph the function and identify any horizontal asymptotes. $$ f(x)=\frac{3 x}{\sqrt{x^{2}+2}} $$

Short answer

Upon graphing the function and observing the trend as x approaches both positive and negative infinity, it can be seen that the function approaches the line \(y=3\) in both directions. Therefore, the horizontal asymptote for the function \(f(x)=\frac{3x}{\sqrt{x^{2}+2}}\) is \(y=3\).

Step-by-step solution

Graph the Function

To graph the function \(f(x)=\frac{3x}{\sqrt{x^{2}+2}}\), plug in various values for x to get the corresponding y values. Plot these points on the graph and connect the dots to make the full graph of function f. Use a graphing utility for a more precise graph.

Identify the Limits as X Approaches Infinity

To identify the horizontal asymptote(s), first find the limit of the function as x approaches positive and negative infinity. You can do this by substituting both \(+\infty\) and \(-\infty\) in place of x in the function. This helps in knowing the value that y is getting closer to as x moves farther along positive or negative direction.

Identify the Horizontal Asymptotes

Look for the line (or lines) that your function is getting closer to as x approaches positive and negative infinity. That line is the horizontal asymptote of the function.