Use the graph of $g(x)=\frac{x^{2}-1}{x^{2}+1}$ and its asymptote to find the following limits.
If necessary, choose the most informative answer from $\infty,-\infty$, and "Does Not Exist."
(a) $\lim _{x \rightarrow \infty} g(x)=\square$
(b) $\lim _{x \rightarrow-\infty} g(x)=\square$
Answer
Final Answer: (b) \(\lim _{x \rightarrow-\infty} g(x)=\boxed{1}\)
Step 1 ：The function \(g(x)=\frac{x^{2}-1}{x^{2}+1}\) is a rational function. The degree of the numerator and the denominator are the same, so the horizontal asymptote of the function is the ratio of the leading coefficients, which is 1.
Step 2 ：Therefore, as \(x\) approaches \(\infty\) or \(-\infty\), the function \(g(x)\) should approach 1.
Step 3 ：Final Answer: (a) \(\lim _{x \rightarrow \infty} g(x)=\boxed{1}\)
Step 4 ：Final Answer: (b) \(\lim _{x \rightarrow-\infty} g(x)=\boxed{1}\)
