Chapter 1

In Exercises 1-10 you are given \(\lim _{x \rightarrow a} f(x)=L\) and a tolerance \(\varepsilon\). Find a number \(\delta\) such that \(|f(x)-L|<\varepsilon\) whenever \(0<|x-a|<\delta\) \(\lim _{x \rightarrow 2} 3 x=6 ; \quad \varepsilon=0.01\)

In Exercises 1-22, find the indicated limit. \(\lim _{t \rightarrow 2}(3 t+4)\)

You are given \(\lim _{x \rightarrow a} f(x)=L\) and a tolerance \(\varepsilon\). Find a number \(\delta\) such that \(|f(x)-L|<\varepsilon\) whenever \(0<|x-a|<\delta\) \(\lim _{x \rightarrow-1} 2 x=-2 ; \quad \varepsilon=0.001\)

Find the indicated limit. \(\lim _{x \rightarrow 2}\left(3 x^{2}+2 x-8\right)\)

You are given \(\lim _{x \rightarrow a} f(x)=L\) and a tolerance \(\varepsilon\). Find a number \(\delta\) such that \(|f(x)-L|<\varepsilon\) whenever \(0<|x-a|<\delta\) \(\lim _{x \rightarrow 1}(2 x+3)=5 ; \quad \varepsilon=0.01\)

Find the indicated limit. \(\lim _{h \rightarrow-1}\left(h^{4}-2 h^{3}+2 h-1\right)\)

You are given \(\lim _{x \rightarrow a} f(x)=L\) and a tolerance \(\varepsilon\). Find a number \(\delta\) such that \(|f(x)-L|<\varepsilon\) whenever \(0<|x-a|<\delta\) \(\lim _{x \rightarrow-2}(3 x-2)=-8 ; \quad \varepsilon=0.05\)

Find the indicated limit. \(\lim _{x \rightarrow 2}\left(x^{2}+1\right)\left(2 x^{2}-4\right)\)

The velocities of car \(A\) and car \(B\), starting out side by side and traveling along a straight road, are given by \(v_{A}=f(t)\) and \(v_{B}=g(t)\), respectively, where \(v\) is measured in feet per second and \(t\) is measured in seconds. a. What can you say about the velocity and acceleration of the two cars at \(t_{1} ?\) (Acceleration is the rate of change of velocity.) b. What can you say about the velocity and acceleration of the two cars at \(t_{2}\) ?

You are given \(\lim _{x \rightarrow a} f(x)=L\) and a tolerance \(\varepsilon\). Find a number \(\delta\) such that \(|f(x)-L|<\varepsilon\) whenever \(0<|x-a|<\delta\) \(\lim _{x \rightarrow 3} \frac{x^{2}-9}{x-3}=6 ; \quad \varepsilon=0.02\)

Find the indicated limit. \(\lim _{x \rightarrow 1}\left(3 x^{2}-4 x+2\right)^{4}\)

Find the indicated limit. \(\lim _{t \rightarrow 3}(2 t-1)^{2}\left(t^{2}-2 t\right)^{3}\)

In Exercises 7-26, find the numbers, if any, where the function is discontinuous. \(f(x)=2 x^{3}-3 x^{2}+4\)

You are given \(\lim _{x \rightarrow a} f(x)=L\) and a tolerance \(\varepsilon\). Find a number \(\delta\) such that \(|f(x)-L|<\varepsilon\) whenever \(0<|x-a|<\delta\) \(\lim _{x \rightarrow 4} \sqrt{x}=2 ; \quad \varepsilon=0.01\)

(a) use Equation (1) to find the slope of the secant line passing through the points \((a, f(a))\) and \((a+h, f(a+h)) ;\) (b) use the results of part (a) and Equafion (2) to find the slope of the tangent line at the point \((a, f(a)) ;\) and \((\mathrm{c})\) find an equation of the tangent line to the graph of \(f\) at the point \((a, f(a))\). \(f(x)=2 x^{2}-1 \quad(2,7)\)

Find the numbers, if any, where the function is discontinuous. \(f(x)=\frac{e^{x}}{x-2}\)

You are given \(\lim _{x \rightarrow a} f(x)=L\) and a tolerance \(\varepsilon\). Find a number \(\delta\) such that \(|f(x)-L|<\varepsilon\) whenever \(0<|x-a|<\delta\) \(\lim _{x \rightarrow 2} \frac{x^{2}+4}{x+2}=2 ; \quad \varepsilon=0.01\)

Find the indicated limit. \(\lim _{x \rightarrow 2}\left(\sqrt{2 x^{3}}-\sqrt{2} x\right)\)

You are given \(\lim _{x \rightarrow a} f(x)=L\) and a tolerance \(\varepsilon\). Find a number \(\delta\) such that \(|f(x)-L|<\varepsilon\) whenever \(0<|x-a|<\delta\) \(\lim _{x \rightarrow 2} \frac{1}{x}=\frac{1}{2} ; \quad \varepsilon=0.05\)

Complete the table by computing \(f(x)\) at the given values of \(x\), accurate to five decimal places. Use the results to guess at the indicated limit, if it exists. $$ \lim _{x \rightarrow 1} \frac{x-1}{x^{2}+x-2} $$

(a) use Equation (1) to find the slope of the secant line passing through the points \((a, f(a))\) and \((a+h, f(a+h)) ;\) (b) use the results of part (a) and Equafion (2) to find the slope of the tangent line at the point \((a, f(a)) ;\) and \((\mathrm{c})\) find an equation of the tangent line to the graph of \(f\) at the point \((a, f(a))\). \(f(x)=x^{3}\) \((2,8)\)

In Exercises 11-22, use the precise definition of a limit to prove that the statement is true. \(\lim _{x \rightarrow 2} 3=3\)

Find the indicated limit. \(\lim _{x \rightarrow-1^{+}}\left(x^{3}-2 x^{2}-5\right)^{2 / 3}\)

Find the numbers, if any, where the function is discontinuous. \(f(x)=\frac{x+1}{x^{2}-2 x-3}\)

Find the indicated limit. \(\lim _{x \rightarrow-2}(x+3)^{2} \sqrt{4 x^{2}-8}\)

Complete the table by computing \(f(x)\) at the given values of \(x\), accurate to five decimal places. Use the results to guess at the indicated limit, if it exists. $$ \lim _{x \rightarrow 0} \frac{\sqrt{3+x}-\sqrt{3-x}}{x} $$

(a) use Equation (1) to find the slope of the secant line passing through the points \((a, f(a))\) and \((a+h, f(a+h)) ;\) (b) use the results of part (a) and Equafion (2) to find the slope of the tangent line at the point \((a, f(a)) ;\) and \((\mathrm{c})\) find an equation of the tangent line to the graph of \(f\) at the point \((a, f(a))\). \(f(x)=\frac{1}{x}\) \((1,1)\)

Use the precise definition of a limit to prove that the statement is true. \(\lim _{x \rightarrow 3} 2 x=6\)

Use the precise definition of a limit to prove that the statement is true. \(\lim _{x \rightarrow-2}(2 x-3)=-7\)

Find the instantaneous rate of change of the given function when \(x=a .\) \(f(x)=2 x^{2}+1 ; \quad a=1\)

Use the precise definition of a limit to prove that the statement is true. \(\lim _{x \rightarrow a} c=c\)

Find the indicated limit. \(\lim _{u \rightarrow-2} \sqrt[3]{\frac{3 u^{2}+2 u}{3 u^{3}-3}}\)

Find the instantaneous rate of change of the given function when \(x=a .\) \(g(x)=x^{2}-x+2 ; \quad a=-1\)

Use the precise definition of a limit to prove that the statement is true. \(\lim _{x \rightarrow a} x=a\)

Find the indicated limit. \(\lim _{w \rightarrow 0} \frac{\sqrt{w+1}-\sqrt{w^{2}+4}}{(w+2)^{2}-(w+1)^{2}}\)

Find the instantaneous rate of change of the given function when \(x=a .\) \(H(x)=x^{3}+x ; \quad a=2\)

Find the numbers, if any, where the function is discontinuous. \(f(x)=x-[x]\)

Use the precise definition of a limit to prove that the statement is true. \(\lim _{x \rightarrow 1} 3 x^{2}=3\)

Find the indicated limit. \(\lim _{x \rightarrow 1} \sin \frac{\pi x}{2}\)

In Exercises 17-22, sketch the graph of the function \(f\) and evaluate (a) \(\lim _{x \rightarrow a^{-}} f(x)\), (b) \(\lim _{x \rightarrow a^{+}} f(x)\), and (c) \(\lim _{x \rightarrow a} f(x)\) for the given value of a. \(f(x)=\left\\{\begin{array}{ll}x-1 & \text { if } x \leq 3 \\ -2 x+8 & \text { if } x>3\end{array} ; \quad a=3\right.\)

Find the instantaneous rate of change of the given function when \(x=a .\) \(f(x)=\sqrt{x} ; \quad a=4\)

Use the precise definition of a limit to prove that the statement is true. \(\lim _{x \rightarrow 2}\left(x^{2}-2\right)=2\)

Find the indicated limit. \(\lim _{x \rightarrow \pi / 4}(x \tan x)\)

Sketch the graph of the function \(f\) and evaluate (a) \(\lim _{x \rightarrow a^{-}} f(x)\), (b) \(\lim _{x \rightarrow a^{+}} f(x)\), and (c) \(\lim _{x \rightarrow a} f(x)\) for the given value of a. \(f(x)=\left\\{\begin{array}{ll}2 x-4 & \text { if } x<4 \\ x-2 & \text { if } x \geq 4\end{array} ; \quad a=4\right.\)

Find the instantaneous rate of change of the given function when \(x=a .\) \(f(x)=\frac{2}{x}+x ; \quad a=1\)

Find the numbers, if any, where the function is discontinuous. \(f(x)=\left\\{\begin{array}{ll}\tan ^{-1}\left|\frac{1}{x-5}\right| & \text { if } x \neq 5 \\ \frac{\pi}{2} & \text { if } x=5\end{array}\right.\)

Use the precise definition of a limit to prove that the statement is true. \(\lim _{x \rightarrow 2} \frac{x^{2}-4}{x-2}=4\)

Find the indicated limit. \(\lim _{x \rightarrow \pi / 4} \frac{\sin x}{x}\)

Sketch the graph of the function \(f\) and evaluate (a) \(\lim _{x \rightarrow a^{-}} f(x)\), (b) \(\lim _{x \rightarrow a^{+}} f(x)\), and (c) \(\lim _{x \rightarrow a} f(x)\) for the given value of a. \(f(x)=\left\\{\begin{array}{ll}-e^{-x} & \text { if } x \neq 0 \\ 1 & \text { if } x=0\end{array} ; a=0\right.\)

Find the instantaneous rate of change of the given function when \(x=a .\) \(f(x)=\frac{1}{x-2} ; \quad a=1\)