91Ó°ÊÓ

Evaluate the limit, if it exists. $$ \lim _{x \rightarrow 3} \frac{\frac{1}{x}-\frac{1}{3}}{x-3} $$

Find the limit or show that it does not exist. $$ \lim _{x \rightarrow \infty} \frac{\sqrt{1+4 x^{6}}}{2-x^{3}} $$

How would you "remove the discontinuity" of \(f ?\) In other words, how would you define \(f(2)\) in order to make \(f\) continuous at \(2 ?\) $$ f(x)=\frac{x^{2}-x-2}{x-2} $$

Use a table of values to estimate the value of the limit. If you have a graphing device, use it to conirm your result graphically. $$ \lim _{x \rightarrow 4} \frac{\ln x-\ln 4}{x-4} $$

Find the derivative of the function using the denition of derivative. State the domain of the function and the domain of its derivative. $$ f(t)=2.5 t^{2}+6 t $$

Sketch the graph of a function \(f\) for which \(f(0)=0,\) \(f^{\prime}(0)=3, f^{\prime}(1)=0,\) and \(f^{\prime}(2)=-1\)

Prove the statement using the \(\varepsilon, \delta\) definition of a limit. $$ \lim _{x \rightarrow a} x=a $$

Find the limit or show that it does not exist. $$ \lim _{x \rightarrow-\infty} \frac{\sqrt{1+4 x^{6}}}{2-x^{3}} $$

Sketch the graph of a function \(g\) for which \(g(0)=g(2)=g(4)=0, g^{\prime}(1)=g^{\prime}(3)=0\) \(g^{\prime}(0)=g^{\prime}(4)=1, g^{\prime}(2)=-1, \lim _{x \rightarrow \infty} g(x)=\infty,\) and \(\lim _{x \rightarrow-\infty} g(x)=-\infty\)

Use a table of values to estimate the value of the limit. If you have a graphing device, use it to conirm your result graphically.$$ \lim _{p \rightarrow-1} \frac{1+p^{9}}{1+p^{15}} $$