91Ó°ÊÓ

A positive number \(\epsilon\) and the limit \(L\) of a function \(f\) at \(-\infty\) are given. Find a negative number \(N\) such that \(|f(x)-L|< \epsilon\) if \(x < N\) $$\lim _{x \rightarrow-\infty} \frac{4 x-1}{2 x+5}=2 ; \epsilon=0.1$$

Given that $$ \lim _{x \rightarrow-\infty} f(x)=0 \quad \text { and } \quad \lim _{x \rightarrow+\infty} f(x)=+\infty $$ evaluate the limit using an appropriate substitution. $$\lim _{x \rightarrow 0^{+}} f(\csc x)$$

In Example 3 we used the Squeezing Theorem to prove that $$\lim _{x \rightarrow 0} x \sin \left(\frac{1}{x}\right)=0$$ Why couldn't we have obtained the same result by writing $$\begin{aligned}\lim _{x \rightarrow 0} x \sin \left(\frac{1}{x}\right) &=\lim _{x \rightarrow 0} x \cdot \lim _{x \rightarrow 0} \sin \left(\frac{1}{x}\right) \\ &=0 \cdot \lim _{x \rightarrow 0} \sin \left(\frac{1}{x}\right)=0 ?\end{aligned}$$

Find the limits. $$\lim _{x \rightarrow-\infty} \frac{1-e^{x}}{1+e^{x}}$$

Writing Suppose that \(f\) and \(g\) are two functions that are equal except at a finite number of points and that \(a\) denotes a real number. Explain informally why both $$ \lim _{x \rightarrow a} f(x) \quad \text { and } \quad \lim _{x \rightarrow a} g(x) $$ exist and are equal, or why both limits fail to exist. Write a short paragraph that explains the relationship of this result to the use of "algebraic simplification" in the evaluation of a limit.

Given that $$ \lim _{x \rightarrow-\infty} f(x)=0 \quad \text { and } \quad \lim _{x \rightarrow+\infty} f(x)=+\infty $$ evaluate the limit using an appropriate substitution. $$\lim _{x \rightarrow 0^{-}} f(\csc x)$$

Find the limits. $$\lim _{x \rightarrow+\infty} \frac{1-e^{x}}{1+e^{x}}$$

Sketch the graphs of the curves \(y=1-x^{2}, y=\cos x\) and \(y=f(x),\) where \(f\) is a function that satisfies the inequalities $$1-x^{2} \leq f(x) \leq \cos x$$ for all \(x\) in the interval \((-\pi / 2, \pi / 2) .\) What can you say about the limit of \(f(x)\) as \(x \rightarrow 0 ?\) Explain.

A positive number \(\epsilon\) and the limit \(L\) of a function \(f\) at \(-\infty\) are given. Find a negative number \(N\) such that \(|f(x)-L|< \epsilon\) if \(x < N\) $$\lim _{x \rightarrow-\infty} \frac{x}{x+1}=1 ; \epsilon=0.001$$

The notion of an asymptote can be extended to include curves as well as lines. Specifically, we say that curves \(y=f(x)\) and \(y=g(x)\) are asymptotic as \(x \rightarrow+\infty\) provided $$ \lim _{x \rightarrow+\infty}[f(x)-g(x)]=0 $$ and are asymptotic as \(x \rightarrow-\infty\) provided $$ \lim _{x \rightarrow-\infty}[f(x)-g(x)]=0 $$ In these exercises, determine a simpler function \(g(x)\) such that \(y=f(x)\) is asymptotic to \(y=g(x)\) as \(x \rightarrow+\infty\) or \(x \rightarrow-\infty\) Use a graphing utility to generate the graphs of \(y=f(x)\) and \(y=g(x)\) and identify all vertical asymptotes. $$\left.f(x)=\frac{x^{2}-2}{x-2} \text { [Hint: Divide } x-2 \text { into } x^{2}-2 .\right]$$