Problem 7
Find the limit (if it exists). If it does not exist, explain why. $$ \lim _{\Delta x \rightarrow 0^{-}} \frac{\frac{1}{x+\Delta x}-\frac{1}{x}}{\Delta x} $$
Problem 7
In Exercises \(7-20,\) find the vertical asymptotes (if any) of the function. $$ h(x)=\frac{x^{2}-2}{x^{2}-x-2} $$
Problem 7
Complete the table and use the result to estimate the limit. Use a graphing utility to graph the function to confirm your result. $$ \begin{aligned} &\lim _{x \rightarrow 0} \frac{\ln (x+1)}{x}\\\ &\begin{array}{|l|l|l|l|l|l|l|} \hline x & -0.1 & -0.01 & -0.001 & 0.001 & 0.01 & 0.1 \\ \hline f(x) & & & & & & \\ \hline \end{array} \end{aligned} $$
Problem 7
Solve for \(x\). $$ 4^{3}=(x+2)^{3} $$
Problem 7
Show that \(f\) and \(g\) are inverse functions (a analytically and (b) graphically. $$ f(x)=1 / x, \quad g(x)=1 / x $$
Problem 7
Find the limit. $$ \lim _{x \rightarrow 7} \frac{5 x}{\sqrt{x+2}} $$
Problem 8
Find the limit. $$ \lim _{x \rightarrow 3} \frac{2 x-5}{x+3} $$
Problem 8
Complete the table and use the result to estimate the limit. Use a graphing utility to graph the function to confirm your result. $$ \begin{aligned} &\lim _{x \rightarrow 2} \frac{\ln x-\ln 2}{x-2}\\\ &\begin{array}{|l|l|l|l|l|l|l|} \hline x & 1.9 & 1.99 & 1.999 & 2.001 & 2.01 & 2.1 \\ \hline f(x) & & & & & & \\ \hline \end{array} \end{aligned} $$
Problem 8
Use the point on the line and the slope of the line to find three additional points that the line passes through. (There is more than one correct answer.) $$ \frac{\text { Point }}{(-2,-2)} \quad \frac{\text { Slope }}{m=2} $$
Problem 8
Solve for \(x\). $$ (x+3)^{4 / 3}=16 $$
