91Ó°ÊÓ

True-False Determine whether the statement is true or false. Explain your answer. Suppose that for all real numbers \(x\), a function \(f\) satisfies $$ |f(x)+5| \leq|x+1| $$ Then \(\lim _{x \rightarrow-1} f(x)=-5\).

True-False Determine whether the statement is true or false. Explain your answer. For \(0

Determine whether the statement is true or false. Explain your answer. If a rational function \(p(x) / q(x)\) has a horizontal asymptote, then the degree of \(p(x)\) must equal the degree of \(q(x)\).

A positive number \(\epsilon\) and the limit \(L\) of a function \(f\) at \(+\infty\) are given. Find a positive number \(N\) such that \(|f(x)-L|<\epsilon\) if \(x>N\). $$ \lim _{x \rightarrow+\infty} \frac{1}{x+2}=0 ; \epsilon=0.005 $$

(a) Explain informally why $$ \lim _{x \rightarrow 0^{-}}\left(\frac{1}{x}+\frac{1}{x^{2}}\right)=+\infty $$ (b) Verify the limit in part (a) algebraically.

Prove: If \(f\) and \(g\) are continuous on \([a, b]\), and \(f(a)>g(a)\), \(f(b)

Let \(p(x)\) and \(q(x)\) be polynomials, with \(q\left(x_{0}\right)=0\). Discuss the behavior of the graph of \(y=p(x) / q(x)\) in the vicinity of \(x=x_{0}\). Give examples to support your conclusions.

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

True-False Determine whether the statement is true or false. Explain your answer. If an invertible function \(f\) is continuous everywhere, then its inverse \(f^{-1}\) is also continuous everywhere.

Give an example of a function \(f\) that is defined on a closed interval, and whose values at the endpoints have opposite signs, but for which the equation \(f(x)=0\) has no solution in the interval.