1.6.3: Additional Exercises

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Reading Questions

What does the statement $|f(x) - L| < \epsilon$ mean in terms of distance?
What does the statement $0 < |x-a| < \delta$ mean in terms of distance and the value of $x$ relative to $a$?
What is a $\delta$-neighborhood centered at $a$? Does it include $a$?
What is an $\epsilon$-neighborhood centered at $L$? Does it include $L$?
State the precise ($\epsilon-\delta$) definition of a finite limit at a finite number: $ \displaystyle \lim_{x \to a} f(x) = L$.
In the $\epsilon-\delta$ definition, which quantity is chosen first, $\epsilon$ or $\delta$? Which one typically depends on the other?
What is the role of "scratch work" in constructing an $\epsilon-\delta$ proof? Which part of the definition (antecedent or consequent) does scratch work typically start with?
When proving a limit for a non-linear function where the $\delta$-neighborhood might not be symmetric (e.g., $f(x)=x^2$ near $x=2$), if you find two possible values $\delta_L$ (from the left) and $\delta_R$ (from the right), how do you choose the final $\delta$?
How is the precise definition of a limit from the right, $ \displaystyle \lim_{x \to a^+} f(x) = L$, different from the definition of a two-sided limit? Specifically, how does the condition on $x-a$ change?
What is the Triangle Inequality?
When proving that a limit does not exist, what needs to be shown regarding $\epsilon$, $\delta$, and $L$?

Homework

In exercises 1 - 4, write the appropriate $ \epsilon − \delta $ definition for each of the given statements.

1) $\displaystyle \lim_{x \to a}f(x)=N$

2) $\displaystyle \lim_{t \to b}g(t)=M$

Answer: For every $ \epsilon >0$, there exists a $ \delta >0$, so that if $0 <|t −b| < \delta $, then $|g(t) −M| < \epsilon $

3) $\displaystyle \lim_{x \to c}h(x)=L$

4) $\displaystyle \lim_{x \to a} \phi (x)=A$

Answer: For every $ \epsilon >0$, there exists a $ \delta >0$, so that if $0 <|x −a| < \delta $, then $| \phi (x) −A| < \epsilon $

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

$A function drawn in quadrant one for x > 0. It is an increasing concave up function, with points approximately (0,0), (1, .5), (2,2), and (3,4).$

5) If $0 <|x −2| < \delta $, then $|f(x) −2| <1$.

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

Answer: $ \delta \leq 0.25$

The following graph of the function $f$ satisfies $\displaystyle \lim_{x \to 3}f(x)= −1$. In the following exercises, determine a value of $ \delta >0$ that satisfies each statement.

$A graph of a decreasing linear function, with points (0,2), (1,1), (2,0), (3,-1), (4,-2), and so on for x >= 0.$

7) If $0 <|x −3| < \delta $, then $|f(x)+1| <1$.

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

Answer: $ \delta \leq 2$

The following graph of the function $f$ satisfies $\displaystyle \lim_{x \to 3}f(x)=2$. In the following exercises, for each value of $ \epsilon $, find a value of $ \delta >0$ such that the precise definition of limit holds true.

$A graph of an increasing linear function intersecting the x axis at about (2.25, 0) and going through the points (3,2) and, approximately, (1,-5) and (4,5).$

9) $ \epsilon =1.5$

10) $ \epsilon =3$

Answer: $ \delta \leq 1$

In exercises 11 - 12, use graphing technology to find a number $ \delta $ such that the statements hold true.

11) $\left|\sin(2x) −\frac{1}{2}\right| <0.1$, whenever $\left|x −\frac{ \pi }{12}\right| < \delta $

12) $\left|\sqrt{x −4} −2\right| <0.1$, whenever $|x −8| < \delta $

Answer: $ \delta <0.3900$

In exercises 13 - 17, use the precise definition of limit to prove the given limits.

13) $\displaystyle \lim_{x \to 2}\,(5x+8)=18$

14) $\displaystyle \lim_{x \to 3}\frac{x^2 −9}{x −3}=6$

Answer: Let $ \delta = \epsilon $. If $0 <|x −3| < \epsilon $, then $\left|\dfrac{x^2 −9}{x −3} - 6\right| = \left|\dfrac{(x+3)(x −3)}{x −3} - 6\right| = |x+3 −6|=|x −3| < \epsilon $.

15) $\displaystyle \lim_{x \to 2}\frac{2x^2 −3x −2}{x −2}=5$

16) $\displaystyle \lim_{x \to 0}x^4=0$

Answer: Let $ \delta =\sqrt[4]{ \epsilon }$. If $0 <|x| <\sqrt[4]{ \epsilon }$, then $\left|x^4-0\right|=x^4 < \epsilon $.

17) $\displaystyle \lim_{x \to 2}\,(x^2+2x)=8$

In exercises 18 - 20, use the precise definition of limit to prove the given one-sided limits.

18) $\displaystyle \lim_{x \to 5^ −}\sqrt{5 −x}=0$

Answer: Let $ \delta = \epsilon ^2$. If $- \epsilon ^2 < x - 5 < 0$, we can multiply through by $-1$ to get $0 <5-x < \epsilon ^2$.
Then $\left|\sqrt{5 −x} - 0\right|=\sqrt{5 −x} < \sqrt{ \epsilon ^2} = \epsilon $.

19) $\displaystyle \lim_{x \to 0^+}f(x)= −2$, where $f(x)=\begin{cases}8x −3, & \text{if }x <0\\4x −2, & \text{if }x \geq 0\end{cases}$.

20) $\displaystyle \lim_{x \to 1^ −}f(x)=3$, where $f(x)=\begin{cases}5x −2, & \text{if }x <1\\7x −1, & \text{if }x \geq 1\end{cases}$.

Answer: Let $ \delta = \epsilon /5$. If $ − \epsilon /5 < x - 1 <0$, we can multiply through by $-1$ to get $0 <1-x < \epsilon /5$.
Then $|f(x) −3|=|5x-2-3| = |5x −5| = 5(1-x)$, since $x <1$ here.
And $5(1-x) < 5( \epsilon /5) = \epsilon $.

21) An engineer is using a machine to cut a flat square of Aerogel of area $144 \,\text{cm}^2$. If there is a maximum error tolerance in the area of $8 \,\text{cm}^2$, how accurately must the engineer cut on the side, assuming all sides have the same length? How do these numbers relate to $ \delta $, $ \epsilon $, $a$, and $L$?

Answer: $0.033 \text{ cm}, \, \epsilon =8,\, \delta =0.33,\,a=12,\,L=144$

22) Use the precise definition of limit to prove that the following limit does not exist: $\displaystyle \lim_{x \to 1}\frac{|x −1|}{x −1}$.

23) Using precise definitions of limits, prove that $\displaystyle \lim_{x \to 0}f(x)$ does not exist, given that $f(x)$ is the ceiling function. (Hint: Try any $ \delta <1$.)

Answer: Answers may very.

24) Using precise definitions of limits, prove that $\displaystyle \lim_{x \to 0}f(x)$ does not exist: $f(x)=\begin{cases}1, & \text{if }x\text{ is rational}\\0, & \text{if }x\text{ is irrational}\end{cases}$. (Hint: Think about how you can always choose a rational number $0 < r < d$, but $ |f(r) - 0| = 1 $.)

25) Using precise definitions of limits, determine $\displaystyle \lim_{x \to 0}f(x)$ for $f(x)=\begin{cases}x, & \text{if }x\text{ is rational}\\0, & \text{if }x\text{ is irrational}\end{cases}$. (Hint: Break into two cases, $x$ rational and $x$ irrational.)

Answer: $0$

26) Using the function from the previous exercise, use the precise definition of limits to show that $\displaystyle \lim_{x \to a}f(x)$ does not exist for $a \neq 0$

For exercises 27 - 29, suppose that $\displaystyle \lim_{x \to a}f(x)=L$ and $\displaystyle \lim_{x \to a}g(x)=M$ both exist. Use the precise definition of limits to prove the following limit laws:

27) $\displaystyle \lim_{x \to a}(f(x) −g(x))=L −M$

Answer: $f(x) −g(x)=f(x)+( −1)g(x)$

28) $\displaystyle \lim_{x \to a}[cf(x)]=cL$ for any real constant $c$ (Hint: Consider two cases: $c=0$ and $c \neq 0$.)

29) $\displaystyle \lim_{x \to a}[f(x)g(x)]=LM$. (Hint: $|f(x)g(x) −LM|= |f(x)g(x) −f(x)M +f(x)M −LM| \leq |f(x)||g(x) −M| +|M||f(x) −L|.)$

Answer: Answers may vary.

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