91Ó°ÊÓ

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 \(x>N\). $$\lim _{x \rightarrow+\infty} \frac{x}{x+1}=1 ; \epsilon=0.001$$

Find all values of \(a\) such that $$ \lim _{x \rightarrow 1}\left(\frac{1}{x-1}-\frac{a}{x^{2}-1}\right) $$ exists and is finite.

If \(y=L\) is a horizontal asymptote for the curve \(y=f(x)\) then it is possible for the graph of \(f\) to intersect the line \(y=L\) infinitely many times.

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

(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)

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 \(x>N\). $$\lim _{x \rightarrow+\infty} \frac{4 x-1}{2 x+5}=2 ; \epsilon=0.1$$

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.

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