91Ó°ÊÓ

(i) Complete the table and make a guess about the limit indicated. (ii) Confirm your conclusions about the limit by graphing a function over an appropriate interval. [Note: For the inverse trigonometric function, be sure to put your calculating and graphing utilities in radian mode. ] $$ f(x)=\frac{\sin ^{-1} 2 x}{x} ; \lim _{x \rightarrow 0} f(x) $$ $$ \begin{array}{|c|c|c|c|c|c|c|}\hline x & {-0.1} & {-0.01} & {-0.001} & {0.001} & {0.01} & {0.1} \\ \hline f(x) & {} & {} & {} & {} \\ \hline\end{array} $$

Find the limits. $$ \lim _{x \rightarrow-\infty} \sqrt{5-x} $$

Find values of \(x,\) if any, at which \(f\) is not continuous. $$ f(x)=\frac{x+2}{x^{2}+4} $$

Find the limits. $$ \lim _{t \rightarrow 2} \frac{t^{3}+3 t^{2}-12 t+4}{t^{3}-4 t} $$

(i) Make a guess at the limit (if it exists) by evaluating the function at the specified \(x\) -values. (ii) Confirm your conclusions about the limit by graphing the function over an appropriate interval. (iii) If you have a CAS, then use it to find the limit. [Note: For the trigonometric functions, be sure to put your calculating and graphing utilities in radian mode.] $$ \begin{array}{l}{\text { (a) } \lim _{x \rightarrow 1} \frac{x-1}{x^{3}-1} ; x=2,1.5,1.1,1.01,1.001,0,0.5,0.9} \\ {\text { (b) } \lim _{x \rightarrow 1^{+}} \frac{x+1}{x^{3}-1} ; x=2,1.5,1.1,1.01,1.001,1.0001} \\ {\text { (c) } \lim _{x \rightarrow 1^{-}} \frac{x+1}{x^{3}-1} ; x=0,0.5,0.9,0.99,0.999,0.9999}\end{array} $$

Find the limits. $$ \lim _{x \rightarrow+\infty} \frac{3 x+1}{2 x-5} $$

(i) Make a guess at the limit (if it exists) by evaluating the function at the specified \(x\) -values. (ii) Confirm your conclusions about the limit by graphing the function over an appropriate interval. (iii) If you have a CAS, then use it to find the limit. [Note: For the trigonometric functions, be sure to put your calculating and graphing utilities in radian mode.] $$ \begin{array}{l}{\text { (a) } \lim _{x \rightarrow 0} \frac{\sqrt{x+1}-1}{x} ; x=\pm 0.25, \pm 0.1, \pm 0.001} \\ {\text { (b) } \lim _{x \rightarrow 0^{+}} \frac{\sqrt{x+1}+1}{x} ; x=0.25,0.1,0.001,0.0001} \\ {\text { (c) } \lim _{x \rightarrow 0^{-}} \frac{\sqrt{x+1}+1}{x} ; x=-0.25,-0.1,-0.001} \\\ {-0.0001}\end{array} $$

Find the limits. $$ \lim _{x \rightarrow+\infty} \frac{5 x^{2}-4 x}{2 x^{2}+3} $$

Find values of \(x,\) if any, at which \(f\) is not continuous. $$ f(x)=\frac{x+2}{x^{2}-4} $$

Find the limits. $$ \lim _{t \rightarrow 1} \frac{t^{3}+t^{2}-5 t+3}{t^{3}-3 t+2} $$