91Ó°ÊÓ

Approximating a Limit Graphically, use a graphing utility to graph the function and approximate the limit accurate to three decimal places. $$ \lim _{x \rightarrow 0} \frac{e^{3 x}-1}{x} $$

Using a Graph to Find a Limit In Exercises 23 and \(24,\) graph the function and find the limit (if it exists) as \(x\) approaches 2 . $$f(x)=\left\\{\begin{array}{cc}{2 x+1,} & {x<2} \\ {x+3,} & {x \geq 2}\end{array}\right.$$

Evaluating a Limit at Infinity In Exercises \(9 - 28\) , find the limit (if it exists). If the limit does not exist, then explain why. Use a graphing utility to verify your result graphically. $$ \lim _ { x \rightarrow \infty } \frac { 5 x ^ { 3 } + 1 } { 10 x ^ { 3 } - 3 x ^ { 2 } + 7 } $$

Sketch a graph of the function and the tangent line at the point \((1, f(1)) .\) Use the graph to approximate the slope of the tangent line. \(f(x)=x^{2}-2\)

Sketch a graph of the function and the tangent line at the point \((1, f(1)) .\) Use the graph to approximate the slope of the tangent line. \(f(x)=x^{2}-2 x+1\)

Approximating a Limit Graphically, use a graphing utility to graph the function and approximate the limit accurate to three decimal places. $$ \lim _{x \rightarrow 0} \frac{1-e^{-x}}{x} $$

Using a Graph to Find a Limit In Exercises 23 and \(24,\) graph the function and find the limit (if it exists) as \(x\) approaches 2 . $$f(x)=\left\\{\begin{array}{ll}{-2 x,} & {x \leq 2} \\ {x^{2}-4 x+1,} & {x>2}\end{array}\right.$$

Evaluating a Limit at Infinity In Exercises \(9 - 28\) , find the limit (if it exists). If the limit does not exist, then explain why. Use a graphing utility to verify your result graphically. $$ \lim _ { x \rightarrow - \infty } \left( \frac { 1 } { 2 } x - \frac { 4 } { x ^ { 2 } } \right) $$

Sketch a graph of the function and the tangent line at the point \((1, f(1)) .\) Use the graph to approximate the slope of the tangent line. \(f(x)=\sqrt{2-x}\)

Approximating a Limit Graphically, use a graphing utility to graph the function and approximate the limit accurate to three decimal places. $$ \lim _{x \rightarrow 0^{+}}(x \ln x) $$