The Limit Laws

In the following exercises (1-2), use the limit laws to evaluate each limit. Justify each step by indicating the appropriate limit law(s).

1. $\underset{x\to 0}{\lim}(4x^2-2x+3)$
2. $\underset{x\to -2}{\lim}\sqrt{x^2-6x+3}$

In the following exercises (3-5), use direct substitution to evaluate each limit.

3. $\underset{x\to 7}{\lim}x^2$
4. $\underset{x\to 0}{\lim}\dfrac{1}{1+ \sin x}$
5. $\underset{x\to 1}{\lim}\dfrac{2-7x}{x+6}$

In the following exercises (6-10), use direct substitution to show that each limit leads to the indeterminate form $\frac{0}{0}$. Then, evaluate the limit.

6. $\underset{x\to 4}{\lim}\dfrac{x^2-16}{x-4}$
7. $\underset{x\to 6}{\lim}\dfrac{3x-18}{2x-12}$
8. $\underset{t\to 9}{\lim}\dfrac{t-9}{\sqrt{t}-3}$
9. $\underset{\theta \to \pi}{\lim}\dfrac{\sin \theta}{\tan \theta}$
    
10. $\underset{x\to 1/2}{\lim}\dfrac{2x^2+3x-2}{2x-1}$

In the following exercises (11-14), assume that $\underset{x\to 6}{\lim}f(x)=4, \, \underset{x\to 6}{\lim}g(x)=9$, and $\underset{x\to 6}{\lim}h(x)=6$. Use these three facts and the limit laws to evaluate each limit.

11. $\underset{x\to 6}{\lim}2f(x)g(x)$
12. $\underset{x\to 6}{\lim}(f(x)+\frac{1}{3}g(x))$
13. $\underset{x\to 6}{\lim}\sqrt{g(x)-f(x)}$
14. $\underset{x\to 6}{\lim}[(x+1)\cdot f(x)]$

In the following exercises (15-16), use a calculator to draw the graph of each piecewise-defined function and study the graph to evaluate the given limits.

15. $f(x)=\begin{cases} x^2, & x \le 3 \\ x+4, & x > 3 \end{cases}$
    1. $\underset{x\to 3^-}{\lim}f(x)$
    2. $\underset{x\to 3^+}{\lim}f(x)$
16. $h(x)=\begin{cases} x^2-2x+1, & x < 2 \\ 3 - x, & x \ge 2 \end{cases}$
    1. $\underset{x\to 2^-}{\lim}h(x)$
    2. $\underset{x\to 2^+}{\lim}h(x)$

In the following exercises (17-20), use the graphs below and the limit laws to evaluate each limit.

17. $\underset{x\to -3^-}{\lim}(f(x)-3g(x))$
18. $\underset{x\to -5}{\lim}\dfrac{2+g(x)}{f(x)}$
19. $\underset{x\to 1}{\lim}\sqrt[3]{f(x)-g(x)}$
20. $\underset{x\to -9}{\lim}[xf(x)+2g(x)]$

Continuity

For the following exercises (1-4), determine the point(s), if any, at which each function is discontinuous. Classify any discontinuity as jump, removable, infinite, or other.

1. $f(x)=\dfrac{1}{\sqrt{x}}$
2. $f(x)=\dfrac{x}{x^2-x}$
3. $f(x)=\dfrac{5}{e^x-2}$
4. $H(x)= \tan 2x$

For the following exercises (5-7), decide if the function continuous at the given point. If it is discontinuous, what type of discontinuity is it?

5. $f(x)=\dfrac{2x^2-5x+3}{x-1}$ at $x=1$
6. $g(u)=\begin{cases} \dfrac{6u^2+u-2}{2u-1} & \text{ if } \, u \ne \frac{1}{2} \\ \dfrac{7}{2} & \text{ if } \, u = \frac{1}{2} \end{cases}$ at $u=\frac{1}{2}$
7. $f(x)=\begin{cases} x^2-e^x & \text{ if } \, x < 0 \\ x-1 & \text{ if } \, x \ge 0 \end{cases}$ at $x=0$

In the following exercises (8-10), find the value(s) of $k$ that makes each function continuous over the given interval.

8. $f(x)=\begin{cases} 3x+2 & \text{ if } \, x < k \\ 2x-3 & \text{ if } \, k \le x \le 8 \end{cases}$
9. $f(x)=\begin{cases} \dfrac{x^2+3x+2}{x+2} & \text{ if } \, x \ne -2 \\ k & \text{ if } \, x = -2 \end{cases}$
10. $f(x)=\begin{cases} \sqrt{kx} & \text{ if } \, 0 \le x \le 3 \\ x+1 & \text{ if } \, 3 < x \le 10 \end{cases}$

In the following exercises (11-12), sketch the graphs.

11. Let $f(x)=\begin{cases} 3x & \text{ if } \, x > 1 \\ x^3 & \text{ if } \, x < 1 \end{cases}$
    1. Sketch the graph of $f$.
    2. Is it possible to find a value $k$ such that $f(1)=k$, which makes $f(x)$ continuous for all real numbers? Briefly explain.
12. Sketch the graph of the function $y=f(x)$ with properties 1 through 7.
    1. The domain of $f$ is $(−\infty,+\infty)$.
    2. $f$ has an infinite discontinuity at $x=-6$.
    3. $f(-6)=3$
    4. $\underset{x\to -3^-}{\lim}f(x)=\underset{x\to -3^+}{\lim}f(x)=2$
    5. $f(-3)=3$
    6. $f$ is left continuous but not right continuous at $x=3$.
    7. $\underset{x\to -\infty}{\lim}f(x)=−\infty$ and $\underset{x\to +\infty}{\lim}f(x)=+\infty$

In the following exercise, suppose $y=f(x)$ is defined for all $x$. For each description, sketch a graph with the indicated property.

13. Discontinuous at $x=1$ with $\underset{x\to -1}{\lim}f(x)=-1$ and $\underset{x\to 2}{\lim}f(x)=4$

Determine whether each of the given statements is true (14-17). Justify your responses with an explanation or counterexample.

14. $f(t)=\dfrac{2}{e^t-e^{-t}}$ is continuous everywhere.
15. If a function is not continuous at a point, then it is not defined at that point.
16. If $f(x)$ is continuous such that $f(a)$ and $f(b)$ have opposite signs, then $f(x)=0$ has exactly one solution in $[a,b]$.
17. If $f(x)$ is continuous everywhere and $f(a), f(b)>0$, then there is no root of $f(x)$ in the interval $[a,b]$.

Prove the following functions are continuous everywhere (18-19).

18. $f(\theta) = \sin \theta$
19. Where is $f(x)=\begin{cases} 0 & \text{ if } \, x \, \text{is irrational} \\ 1 & \text{ if } \, x \, \text{is rational} \end{cases}$ continuous?

The Precise Definition of a Limit

In the following exercises (1-2), write the appropriate $\varepsilon$–$\delta$ definition for each of the given statements.

1. $\underset{t\to b}{\lim}g(t)=M$
2. $\underset{x\to a}{\lim}\phi(x)=A$

The following graph of the function $f$ satisfies $\underset{x\to 2}{\lim}f(x)=2$. In the following exercise, determine a value of $\delta >0$ that satisfies the statement.

3. If $0<|x-2|<\delta$, then $|f(x)-2|<0.5$.

The following graph of the function $f$ satisfies $\underset{x\to 3}{\lim}f(x)=-1$. In the following exercise, determine a value of $\delta >0$ that satisfies the statement.

4. If $0<|x-3|<\delta$, then $|f(x)+1|<2$.

The following graph of the function $f$ satisfies $\underset{x\to 3}{\lim}f(x)=2$. In the following exercise, for the given value of $\varepsilon$, find a value of $\delta >0$ such that the precise definition of limit holds true.

5. $\varepsilon =3$

In the following exercise, use a graphing calculator to find a number $\delta$ such that the statement holds true.

6. $|\sqrt{x-4}-2|<0.1$, whenever $|x-8|<\delta$

In the following exercises (7-8), use the precise definition of limit to prove the given limits.

7. $\underset{x\to 3}{\lim}\dfrac{x^2-9}{x-3}=6$
8. $\underset{x\to 0}{\lim}x^4=0$

In the following exercises (9-10), use the precise definition of limit to prove the given one-sided limits.

9. $\underset{x\to 5^-}{\lim}\sqrt{5-x}=0$
10. $\underset{x\to 1^-}{\lim}f(x)=3$, where $f(x)=\begin{cases} 5x-2 & \text{ if } \, x < 1 \\ 7x-1 & \text{ if } x \ge 1 \end{cases}$

In the following exercise, use the precise definition of limit to prove the given infinite limit.

11. $\underset{x\to -1}{\lim}\dfrac{3}{(x+1)^2}=\infty$

For the following exercises (12-13), suppose that $\underset{x\to a}{\lim}f(x)=L$ and $\underset{x\to a}{\lim}g(x)=M$ both exist. Use the precise definition of limits to prove the following limit laws:

12. $\underset{x\to a}{\lim}(f(x)-g(x))=L-M$
13. $\underset{x\to a}{\lim}[f(x)g(x)]=LM$.