91Ó°ÊÓ

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 \(\lim _{x \rightarrow-\infty} g(x)=-\infty\)

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

Find the derivative of the function using the denition of derivative. State the domain of the function and the domain of its derivative. $$ f(x)=4+8 x-5 x^{2} $$

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

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

Prove the statement using the \(\varepsilon, \delta\) definition of a limit. $$ \lim _{x \rightarrow 0} x^{2}=0 $$

Find the derivative of the function using the denition of derivative. State the domain of the function and the domain of its derivative. $$ f(x)=x^{2}-2 x^{3} $$

Sketch the graph of a function \(g\) that is continuous on its domain \((-5,5)\) and where \(g(0)=1, g^{\prime}(0)=1, g^{\prime}(-2)=0\) \(\lim _{x \rightarrow-5^{+}} g(x)=\infty,\) and \(\lim _{x \rightarrow 5^{-}} g(x)=3\)

Evaluate the limit, if it exists. $$ \lim _{t \rightarrow 0} \frac{\sqrt{1+t}-\sqrt{1-t}}{t} $$

Prove the statement using the \(\varepsilon, \delta\) definition of a limit. $$ \lim _{x \rightarrow 0} x^{3}=0 $$