91Ó°ÊÓ

HOW DO YOU SEE IT? For the function $$ f(x)=\left\\{\begin{array}{ll}{2 x,} & {x \leq 0} \\ {x^{2}+1,} & {x>0}\end{array}\right. $$ shown at the right, find each of the following limits. If the limit does not exist, then explain why. $$ \begin{array}{l}{\text { (a) } \lim _{x \rightarrow 0^{-}} f(x)} \\ {\text { (b) } \lim _{x \rightarrow 0^{+}} f(x)} \\ {\text { (c) } \lim _{x \rightarrow 0} f(x)}\end{array} $$

Evaluating a Limit by Direct Substitution Exercises \(45-64\) , find the limit by direct substitution. $$\lim _{x \rightarrow 3} \sqrt[3]{x^{2}-1}$$

Finding the Limit of a Sequence In Exercises \(55 - 58\) , find the limit of the sequence. Then verify the limit numerically by using a graphing utility to complete the table. $$\begin{array} { | c | c | c | c | c | c | c | } \hline n & { 10 ^ { 0 } } & { 10 ^ { 1 } } & { 10 ^ { 2 } } & { 10 ^ { 3 } } & { 10 ^ { 4 } } & { 10 ^ { 5 } } & { 10 ^ { 6 } } \\ \hline a _ { n } & { } & { } & { } & { } & { } & { } \\\ \hline \end{array}$$ $$ a _ { n } = \frac { 4 } { n } \left( n + \frac { 4 } { n } \left[ \frac { n ( n + 1 ) } { 2 } \right] \right) $$

True or False?, determine whether the statement is true or false. Justify your answer. When your attempt to find the limit of a rational function yields the indeterminate form \(\frac{0}{0},\) the rational function's numerator and denominator have a common factor.

Find an equation of the line that is tangent to the graph of \(f\) and parallel to the given line. Function \(\quad\) Line \(f(x)=-\frac{1}{2} x^{3} \quad 6 x+y+4=0\)

Finding the Limit of a Sequence In Exercises \(55 - 58\) , find the limit of the sequence. Then verify the limit numerically by using a graphing utility to complete the table. $$\begin{array} { | c | c | c | c | c | c | c | } \hline n & { 10 ^ { 0 } } & { 10 ^ { 1 } } & { 10 ^ { 2 } } & { 10 ^ { 3 } } & { 10 ^ { 4 } } & { 10 ^ { 5 } } & { 10 ^ { 6 } } \\ \hline a _ { n } & { } & { } & { } & { } & { } & { } \\\ \hline \end{array}$$ $$ a _ { n } = \frac { 16 } { n ^ { 3 } } \left[ \frac { n ( n + 1 ) ( 2 n + 1 ) } { 6 } \right] $$

Evaluating a Limit by Direct Substitution Exercises \(45-64\) , find the limit by direct substitution. $$\lim _{x \rightarrow 7} \frac{5 x}{\sqrt{x+2}}$$

True or False?, determine whether the statement is true or false. Justify your answer. If $$ f(c)=L, \text { then } \lim _{x \rightarrow c} f(x)=L $$

Find an equation of the line that is tangent to the graph of \(f\) and parallel to the given line. Function \(\quad\) Line \(f(x)=x^{2}-x \quad x+2 y-6=0\)

Finding the Limit of a Sequence In Exercises \(55 - 58\) , find the limit of the sequence. Then verify the limit numerically by using a graphing utility to complete the table. $$\begin{array} { | c | c | c | c | c | c | c | } \hline n & { 10 ^ { 0 } } & { 10 ^ { 1 } } & { 10 ^ { 2 } } & { 10 ^ { 3 } } & { 10 ^ { 4 } } & { 10 ^ { 5 } } & { 10 ^ { 6 } } \\ \hline a _ { n } & { } & { } & { } & { } & { } & { } \\\ \hline \end{array}$$ $$ a _ { n } = \frac { n ( n + 1 ) } { n ^ { 2 } } - \frac { 1 } { n ^ { 4 } } \left[ \frac { n ( n + 1 ) } { 2 } \right] ^ { 2 } $$