Question

In Exercises 59 and \(60,\) (a) explain why L'Hôpital's Rule cannot be used to find the limit, (b) find the limit analytically, and (c) use a graphing utility to graph the function and approximate the limit from the graph. Compare the result with that in part (b). \(\lim _{x \rightarrow \infty} \frac{x}{\sqrt{x^{2}+1}}\)

Short answer

The limit as \(x\) goes to infinity of \( \frac{x}{\sqrt{x^{2}+1}}\) is 1.

Step-by-step solution

Rewrite the Limit

Rewrite the limit \(\lim_{x \rightarrow \infty} \frac{x}{\sqrt{x^{2}+1}}\) by factoring out \(x\) from the radical in the denominator.

Simplify the Limit

When you factor out \(x\) from the root, you get the expression \( \lim_{x \rightarrow \infty} \frac{x}{x\sqrt{1+\frac{1}{x^{2}}}}\). After simplifying, you get \( \lim_{x \rightarrow \infty} \frac{1}{\sqrt{1+\frac{1}{x^2}}}\)

Evaluate the Limit

As \(x\) goes to infinity, the expression \(\frac{1}{x^{2}}\) tends to 0, which simplifies the limit to \( \lim_{x \rightarrow \infty} \frac{1}{\sqrt{1}}\). Hence, the limit of the given function as \(x\) goes to infinity is 1.

Graph the Function

Graph the function \( \frac{x}{\sqrt{x^{2}+1}}\) and observe its behaviour as \(x\) approaches to infinity, which indeed confirms the calculated limit.