Chapter 2 Review Exercises

Chapter Review Exercises

True or False. In the following exercises, justify your answer with a proof or a counterexample.

A function has to be continuous at [latex]x=a[/latex] if the [latex]\underset{x\to a}{\text{lim}}f(x)[/latex] exists.

You can use the quotient rule to evaluate [latex]\underset{x\to 0}{\text{lim}}\frac{ \sin x}{x}.[/latex]

If there is a vertical asymptote at [latex]x=a[/latex] for the function [latex]f(x),[/latex] then [latex]f[/latex] is undefined at the point [latex]x=a.[/latex]

If [latex]\underset{x\to a}{\text{lim}}f(x)[/latex] does not exist, then [latex]f[/latex] is undefined at the point [latex]x=a.[/latex]

Using the graph, find each limit or explain why the limit does not exist.

  1. [latex]\underset{x\to -1}{\text{lim}}f(x)[/latex]
  2. [latex]\underset{x\to 1}{\text{lim}}f(x)[/latex]
  3. [latex]\underset{x\to {0}^{+}}{\text{lim}}f(x)[/latex]
  4. [latex]\underset{x\to 2}{\text{lim}}f(x)[/latex]

A graph of a piecewise function with several segments. The first is a decreasing concave up curve existing for x < -1. It ends at an open circle at (-1, 1). The second is an increasing linear function starting at (-1, -2) and ending at (0,-1). The third is an increasing concave down curve existing from an open circle at (0,0) to an open circle at (1,1). The fourth is a closed circle at (1,-1). The fifth is a line with no slope existing for x > 1, starting at the open circle at (1,1).

In the following exercises, evaluate the limit algebraically or explain why the limit does not exist.

[latex]\underset{x\to 2}{\text{lim}}\frac{2{x}^{2}-3x-2}{x-2}[/latex]

[latex]\underset{x\to 0}{\text{lim}}3{x}^{2}-2x+4[/latex]

[latex]\underset{x\to 3}{\text{lim}}\frac{{x}^{3}-2{x}^{2}-1}{3x-2}[/latex]

[latex]\underset{x\to \pi \text{/}2}{\text{lim}}\frac{ \cot x}{ \cos x}[/latex]

[latex]\underset{x\to -5}{\text{lim}}\frac{{x}^{2}+25}{x+5}[/latex]

[latex]\underset{x\to 2}{\text{lim}}\frac{3{x}^{2}-2x-8}{{x}^{2}-4}[/latex]

[latex]\underset{x\to 1}{\text{lim}}\frac{{x}^{2}-1}{{x}^{3}-1}[/latex]

[latex]\underset{x\to 1}{\text{lim}}\frac{{x}^{2}-1}{\sqrt{x}-1}[/latex]

[latex]\underset{x\to 4}{\text{lim}}\frac{4-x}{\sqrt{x}-2}[/latex]

[latex]\underset{x\to 4}{\text{lim}}\frac{1}{\sqrt{x}-2}[/latex]

In the following exercises, use the squeeze theorem to prove the limit.

[latex]\underset{x\to 0}{\text{lim}}{x}^{2} \cos (2\pi x)=0[/latex]

[latex]\underset{x\to 0}{\text{lim}}{x}^{3} \sin (\frac{\pi }{x})=0[/latex]

Determine the domain such that the function [latex]f(x)=\sqrt{x-2}+x{e}^{x}[/latex] is continuous over its domain.

In the following exercises, determine the value of [latex]c[/latex] such that the function remains continuous. Draw your resulting function to ensure it is continuous.

[latex]f(x)=\bigg\{\begin{array}{l}{x}^{2}+1,x>c\\ 2x,x\le c\end{array}[/latex]

[latex]f(x)=\bigg\{\begin{array}{l}\sqrt{x+1},x>\text{−}1\\ {x}^{2}+c,x\le -1\end{array}[/latex]

In the following exercises, use the precise definition of limit to prove the limit.

[latex]\underset{x\to 1}{\text{lim}}(8x+16)=24[/latex]

[latex]\underset{x\to 0}{\text{lim}}{x}^{3}=0[/latex]

A ball is thrown into the air and the vertical position is given by [latex]x(t)=-4.9{t}^{2}+25t+5.[/latex] Use the Intermediate Value Theorem to show that the ball must land on the ground sometime between 5 sec and 6 sec after the throw.

A particle moving along a line has a displacement according to the function [latex]x(t)={t}^{2}-2t+4,[/latex] where [latex]x[/latex] is measured in meters and [latex]t[/latex] is measured in seconds. Find the average velocity over the time period [latex]t=\left[0,2\right].[/latex]

From the previous exercises, estimate the instantaneous velocity at [latex]t=2[/latex] by checking the average velocity within [latex]t=0.01 \sec \text{.}[/latex]