2.3E: Exercises

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Reading Questions

What are the two Basic Limit Laws for $ \displaystyle \lim_{x \to a} x$ and $ \displaystyle \lim_{x \to a} c$?
State the Sum Law for Limits. What condition must be met for it to apply?
Under what condition can the Quotient Law for Limits be applied?
When can the Direct Substitution Property be used to evaluate the limit of a polynomial function?
When can the Direct Substitution Property be used to evaluate the limit of a rational function $\frac{p(x)}{q(x)}$ at $x=a$? What must be true about $q(a)$?
If direct substitution into a rational function yields $\frac{0}{0}$, what kind of algebraic manipulation might be useful before re-evaluating the limit?
How do the Limit Laws apply to one-sided limits? What condition must be met for the function's domain?
Explain the Squeeze Theorem in your own words. What three conditions must be met to use it to find $ \displaystyle \lim_{x \to a} g(x)$?
What is the value of $ \displaystyle \lim_{\theta \to 0} \sin(\theta)$?
What is the value of $ \displaystyle \lim_{\theta \to 0} \cos(\theta)$?
What is the important limit $ \displaystyle \lim_{\theta \to 0} \frac{\sin\theta}{\theta}$?
What is the important limit $ \displaystyle \lim_{\theta \to 0} \frac{1-\cos\theta}{\theta}$?
Why is it crucial to use radian measure (not degrees) when working with limits of trigonometric functions in calculus?
What does it mean if applying Limit Laws leads to an indeterminate form like $0 \cdot \infty$? What should be done instead?

Homework

In exercises 1 - 4, use the Limit Laws to evaluate each limit. Justify each step by indicating the appropriate Limit Law(s).

1) $\displaystyle \lim_{x \to 0}\,(4x^2−2x+3)$

2) $\displaystyle \lim_{x \to 1}\frac{x^3+3x^2+5}{4−7x}$

3) $\displaystyle \lim_{x \to −2}\sqrt{x^2−6x+3}$

4) $\displaystyle \lim_{x \to −1}(9x+1)^2$

In exercises 5 - 8, use the Direct Substitution Property to evaluate the limit of each function.

5) $\displaystyle \lim_{x \to 7}x^2$

6) $\displaystyle \lim_{x \to −2}(4x^2−1)$

7) $\displaystyle \lim_{x \to 0}\frac{1}{1+\sin x}$

8) $\displaystyle \lim_{x \to 1}\frac{2−7x}{x+6}$

In exercises 9 - 16, assume that $\displaystyle \lim_{x \to 6}f(x)=4$, $\displaystyle \lim_{x \to 6}g(x)=9$, and $\displaystyle \lim_{x \to 6}h(x)=6$. Use these three facts and the Limit Laws to evaluate each limit.

9) $\displaystyle \lim_{x \to 6}2f(x)g(x)$

10) $\displaystyle \lim_{x \to 6}\frac{g(x)−1}{f(x)}$

11) $\displaystyle \lim_{x \to 6}\left(f(x)+\frac{1}{3}g(x)\right)$

12) $\displaystyle \lim_{x \to 6}\frac{\big(h(x)\big)^3}{2}$

13) $\displaystyle \lim_{x \to 6}\sqrt{g(x)−f(x)}$

14) $\displaystyle \lim_{x \to 6}x \cdot h(x)$

15) $\displaystyle \lim_{x \to 6}[(x+1) \cdot f(x)]$

16) $\displaystyle \lim_{x \to 6}(f(x) \cdot g(x)−h(x))$

In exercises 17 - 19, evaluate the limits using the given piecewise-defined function and the Limit Laws.

17) $f(x)=\begin{cases}x^2, & x \leq 3\\ x+4, & x>3\end{cases}$

a. $\displaystyle \lim_{x \to 3^−}f(x)$

b. $\displaystyle \lim_{x \to 3^+}f(x)$

18) $g(x)=\begin{cases}x^3−1, & x \leq 0\\1, & x>0\end{cases}$

a. $\displaystyle \lim_{x \to 0^−}g(x)$

b. $\displaystyle \lim_{x \to 0^+}g(x)$

19) $h(x)=\begin{cases}x^2−2x+1, & x<2\\3−x, & x \geq 2\end{cases}$

a. $\displaystyle \lim_{x \to 2^−}h(x)$

b. $\displaystyle \lim_{x \to 2^+}h(x)$

In exercises 20 - 27, use the following graphs and the Limit Laws to evaluate each limit.

$Two graphs of piecewise functions. The upper is f(x), which has two linear segments. The first is a line with negative slope existing for x < -3. It goes toward the point (-3,0) at x= -3. The next has increasing slope and goes to the point (-3,-2) at x=-3. It exists for x > -3. Other key points are (0, 1), (-5,2), (1,2), (-7, 4), and (-9,6). The lower piecewise function has a linear segment and a curved segment. The linear segment exists for x < -3 and has decreasing slope. It goes to (-3,-2) at x=-3. The curved segment appears to be the right half of a downward opening parabola. It goes to the vertex point (-3,2) at x=-3. It crosses the y axis a little below y=-2. Other key points are (0, -7/3), (-5,0), (1,-5), (-7, 2), and (-9, 4).$

20) $\displaystyle \lim_{x \to −3^+}(f(x)+g(x))$

21) $\displaystyle \lim_{x \to −3^−}(f(x)−3g(x))$

22) $\displaystyle \lim_{x \to 0}\frac{f(x)g(x)}{3}$

23) $\displaystyle \lim_{x \to −5}\frac{2+g(x)}{f(x)}$

24) $\displaystyle \lim_{x \to 1}(f(x))^2$

25) $\displaystyle \lim_{x \to 1}\sqrt[3]{f(x)−g(x)}$

26) $\displaystyle \lim_{x \to −7}(x \cdot g(x))$

27) $\displaystyle \lim_{x \to −9}[x \cdot f(x)+2 \cdot g(x)]$

For exercises 28 - 30, evaluate the limit using the Squeeze Theorem. Graph the functions $f(x)$, $g(x)$, and $h(x)$ when possible.

28) True or False? If $2x−1 \leq g(x) \leq x^2−2x+3$, then $\displaystyle \lim_{x \to 2}g(x)=0$.

29) $\displaystyle \lim_{ \theta \to 0} \theta ^2\cos\left(\frac{1}{ \theta }\right)$

30) $\displaystyle \lim_{x \to 0}f(x)$, where $f(x)=\begin{cases}0, & x\text{ rational}\\ x^2, & x\text{ irrrational}\end{cases}$

31) In physics, the magnitude of an electric field generated by a point charge at a distance $r$ in vacuum is governed by Coulomb’s law: $E(r)=\dfrac{q}{4 \pi \epsilon _0r^2}$, where $E$ represents the magnitude of the electric field, $q$ is the charge of the particle, $r$ is the distance between the particle and where the strength of the field is measured, and $\dfrac{1}{4 \pi \epsilon _0}$ is Coulomb’s constant: $8.988 \times 109N \cdot m^2/C^2$.

a. Use a graphing calculator to graph $E(r)$ given that the charge of the particle is $q=10^{−10}$.

b. Evaluate $\displaystyle \lim_{r \to 0^+}E(r)$. What is the physical meaning of this quantity? Is it physically relevant? Why are you evaluating from the right?

32) The density of an object is given by its mass divided by its volume: $ \rho =m/V$.

a. Use a calculator to plot the volume as a function of density $(V=m/ \rho )$, assuming you are examining something of mass $8$ kg ($m=8$).

b. Evaluate $\displaystyle \lim_{x \to 0^+}V(\rho)$ and explain the physical meaning.

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