91Ó°ÊÓ

Find the limit or show that it does not exist. $$ \lim _{x \rightarrow \infty}(\sqrt{x^{2}+a x}-\sqrt{x^{2}+b x}) $$

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

(a) If \(G(x)=4 x^{2}-x^{3}\), find \(G^{\prime}(a)\) and use it to find equa- tions of the tangent lines to the curve \(y=4 x^{2}-x^{3}\) at the points \((2,8)\) and \((3,9) .\) (b) Illustrate part (a) by graphing the curve and the tangent lines on the same screen.

Evaluate the limit, if it exists. $$ \lim _{x \rightarrow-4} \frac{\sqrt{x^{2}+9}-5}{x+4} $$

Prove the statement using the \(\varepsilon, \delta\) definition of a limit. $$ \lim _{x \rightarrow 2}\left(x^{2}+2 x-7\right)=1 $$

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

(a) Estimate the value of $$ \lim _{x \rightarrow 0} \frac{\sin x}{\sin \pi x} $$ by graphing the function \(f(x)=(\sin x) /(\sin \pi x)\) State your answer correct to two decimal places. (b) Check your answer in part (a) by evaluating \(f(x)\) for values of \(x\) that approach \(0 .\)

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

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

Prove the statement using the \(\varepsilon, \delta\) definition of a limit. $$ \lim _{x \rightarrow-2}\left(x^{2}-1\right)=3 $$