91Ó°ÊÓ

Make a careful sketch of the graph of \(f\) and below it sketch the graph of \(f^{\prime}\) in the same manner as in Exercises \(4-11 .\) Can you guess a formula for \(f^{\prime}(x)\) from its graph? $$ f(x)=\sin x $$

Evaluate the limit, if it exists. $$ \lim _{x \rightarrow-1} \frac{2 x^{2}+3 x+1}{x^{2}-2 x-3} $$

Use the definition of continuity and the properties of limits to show that the function is continuous on the given interval. $$ g(x)=\frac{x-1}{3 x+6}, \quad(-\infty,-2) $$

Evaluate the limit, if it exists. $$ \lim _{h \rightarrow 0} \frac{(-5+h)^{2}-25}{h} $$

Make a careful sketch of the graph of \(f\) and below it sketch the graph of \(f^{\prime}\) in the same manner as in Exercises \(4-11 .\) Can you guess a formula for \(f^{\prime}(x)\) from its graph? $$ f(x)=e^{x} $$

Prove the statement using the \(\varepsilon, \delta\) definition of a limit and illustrate with a diagram like Figure \(9 .\) $$ \lim _{x \rightarrow-3}(1-4 x)=13 $$

Sketch the graph of an example of a function \(f\) that satisfies all of the given conditions. $$ \begin{array}{l}{\lim _{x \rightarrow 3^{+}} f(x)=4, \quad \lim _{x \rightarrow 3^{-}} f(x)=2, \quad \lim _{x \rightarrow-2} f(x)=2} \\ {f(3)=3, \quad f(-2)=1}\end{array} $$

Explain why the function is discontinuous at the given number \(a\). Sketch the graph of the function. $$ f(x)=\frac{1}{x+2} \quad \quad a=-2 $$

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

Evaluate the limit, if it exists. $$ \lim _{h \rightarrow 0} \frac{(2+h)^{3}-8}{h} $$