91Ó°ÊÓ

Estimate the limits numerically. \(\lim _{x \rightarrow 2} e^{x-2}\)

Use a graph of for some other method to determine what, if any, value to assign to \(f(a)\) to make \(f\) continuous at \(x=a\). $$ f(x)=\frac{x^{2}-2 x+1}{x-1} ; a=1 $$

In Exercises 15-28, compute the derivative function \(f^{\prime}(x)\) algebraically. (Notice that the functions are the same as those in Exercises \(1-14 .)\) HINT [See Examples 2 and \(3 .]\) $$ f(x)=x^{2}+1 $$

Determine if the given limit leads to a determinate or indeterminate form. Evaluate the limit if it exists, or say why if not. $$ \lim _{x \rightarrow-\infty} \frac{-x^{3}}{3 x^{6}} $$

Compute the derivative function \(f^{\prime}(x)\) algebraically. (Notice that the functions are the same as those in Exercises \(1-14 .)\) HINT [See Examples 2 and \(3 .]\) $$ f(x)=x^{2}-3 $$

Calculate the average rate of change of the given function over the given interval. Where appropriate, specify the units of measurement. HINT [See Example 1.] $$ f(x)=\frac{1}{x} ;[1,4] $$

Determine if the given limit leads to a determinate or indeterminate form. Evaluate the limit if it exists, or say why if not. $$ \lim _{x \rightarrow+\infty} \frac{-x^{6}}{3 x^{3}} $$

Estimate the limits numerically. \(\lim _{x \rightarrow+\infty} e^{-x}\)

Calculate the average rate of change of the given function over the given interval. Where appropriate, specify the units of measurement. HINT [See Example 1.] $$ f(x)=\frac{x^{2}}{2}+\frac{1}{x} ;[2,3] $$

Compute the derivative function \(f^{\prime}(x)\) algebraically. (Notice that the functions are the same as those in Exercises \(1-14 .)\) HINT [See Examples 2 and \(3 .]\) $$ f(x)=3 x-4 $$