Problem 3
Find the limit. $$ \lim _{x \rightarrow 2} x^{4} $$
Problem 3
Sketch the lines through the given point with the indicated slopes. Make the sketches on the same set of coordinate axes. $$ \begin{array}{llllll} \text { 3. } \frac{\text { Point }}{(2,3)} & \text { Slopes } & \\ \hline \text { a) } 1 & \text { (b) }-2 & \text { (c) }-\frac{3}{2} & \text { (d) Undefined } \end{array} $$
Problem 3
Numerical and Graphical Analysis In Exercises 3-6, determine whether \(f(x)\) approaches \(\infty\) or \(-\infty\) as \(x\) approaches -3 from the left and from the right by completing the table. Use a graphing utility to graph the function to confirm your answer. $$ f(x)=\frac{1}{x^{2}-9} $$
Problem 3
Complete the table and use the result to estimate the limit. Use a graphing utility to graph the function to confirm your result. $$ \begin{aligned} &\lim _{x \rightarrow 0} \frac{\sin x}{x}\\\ &\begin{array}{|l|l|l|l|l|l|l|} \hline x & -0.1 & -0.01 & -0.001 & 0.001 & 0.01 & 0.1 \\ \hline f(x) & & & & & & \\ \hline \end{array} \end{aligned} $$
Problem 3
Use the properties of exponents to simplify the expressions. (a) \(\left(5^{2}\right)\left(5^{3}\right)\) (b) \(\left(5^{2}\right)\left(5^{-3}\right)\) (c) \(\frac{5^{3}}{25^{2}}\) (d) \(\left(\frac{1}{4}\right)^{2} 2^{6}\)
Problem 3
Show that \(f\) and \(g\) are inverse functions (a analytically and (b) graphically. $$ f(x)=x^{3}, \quad g(x)=\sqrt[3]{x} $$
Problem 3
Evaluate (if possible) the function at the given value(s) of the independent variable. Simplify the results. \(g(x)=3-x^{2}\) (a) \(g(0)\) (b) \(g(\sqrt{3})\) (c) \(g(-2)\) (d) \(g(t-1)\)
Problem 4
Numerical and Graphical Analysis In Exercises 3-6, determine whether \(f(x)\) approaches \(\infty\) or \(-\infty\) as \(x\) approaches -3 from the left and from the right by completing the table. Use a graphing utility to graph the function to confirm your answer. $$ f(x)=\frac{x}{x^{2}-9} $$
Problem 4
Use the properties of exponents to simplify the expressions. (a) \(\left(\frac{1}{e}\right)^{-2}\) (b) \(\left(\frac{e^{5}}{e^{2}}\right)^{-1}\) (c) \(e^{0}\) (d) \(\frac{1}{e^{-3}}\)
Problem 4
Evaluate (if possible) the function at the given value(s) of the independent variable. Simplify the results. \(f(x)=\sqrt{x+3}\) (a) \(f(-2)\) (b) \(f(6)\) (c) \(f(-5)\) (d) \(f(x+\Delta x)\)
