2.5E: Exercises for Section 2.5

Exercises 1 - 3

Write the appropriate \( ε − δ\) definition for each of the given statements.

1. \(\displaystyle \lim_{t →b}g(t)=M\)

Answer

For every \( ε >0\), there exists a \( δ >0\), so that if \(0 <|t −b| < δ\), then \(|g(t) −M| < ε\)

2. \(\displaystyle \lim_{x →c}h(x)=L\)
3. \(\displaystyle \lim_{x →a} φ(x)=A\)

Answer

For every \( ε >0\), there exists a \( δ >0\), so that if \(0 <|x −a| < δ\), then \(| φ(x) −A| < ε\)

Exercises 4 - 5

The following graph of the function \(f\) in Figure \(\PageIndex{1}\) satisfies \(\displaystyle \lim_{x →2}f(x)=2\). In the following exercises, determine a value of \( δ >0\) that satisfies each statement.

4. If \(0 <|x −2| < δ\), then \(|f(x) −2| <1\).
5. If \(0 <|x −2| < δ\), then \(|f(x) −2| <0.5\).

Answer

\( δ ≤0.25\)

Exercises 6 - 7

The following graph of the function \(f\) in Figure \(\PageIndex{2}\) satisfies \(\displaystyle \lim_{x →3}f(x)= −1\). In the following exercises, determine a value of \( δ >0\) that satisfies each statement.

6. If \(0 <|x −3| < δ\), then \(|f(x)+1| <1\).
7. If \(0 <|x −3| < δ\), then \(|f(x)+1| <2\).

Answer

\( δ ≤2\)

Exercises 8 - 9

The following graph of the function \(f\) in Figure \(\PageIndex{3}\) satisfies \(\displaystyle \lim_{x →3}f(x)=2\). In the following exercises, for each value of \( ε\), find a value of \( δ >0\) such that the precise definition of limit holds true.

8. \( ε=1.5\)
9. \( ε=3\)

Answer

\( δ ≤1\)

[T] Exercises 10 - 11

Use a graphing calculator to find a number \( δ\) such that the statements hold true.

10. \(\left|\sin(2x) −\frac{1}{2}\right| <0.1\), whenever \(\left|x −\frac{ π}{12}\right| < δ\)
11. \(\left|\sqrt{x −4} −2\right| <0.1\), whenever \(|x −8| < δ\)

Answer

\( δ <0.3900\)

Exercises 12 - 16

Use the precise definition of limit to prove the given limits.

12. \(\displaystyle \lim_{x →2}\,(5x+8)=18\)
13. \(\displaystyle \lim_{x →3}\frac{x^2 −9}{x −3}=6\)

Answer

Let \( δ= ε\). If \(0 <|x −3| < ε\), then \(\left|\dfrac{x^2 −9}{x −3} - 6\right| = \left|\dfrac{(x+3)(x −3)}{x −3} - 6\right| = |x+3 −6|=|x −3| < ε\).

14. \(\displaystyle \lim_{x →2}\frac{2x^2 −3x −2}{x −2}=5\)
15. \(\displaystyle \lim_{x →0}x^4=0\)

Answer

Let \( δ=\sqrt[4]{ ε}\). If \(0 <|x| <\sqrt[4]{ ε}\), then \(\left|x^4-0\right|=x^4 < ε\).

16. \(\displaystyle \lim_{x →2}\,(x^2+2x)=8\)

Exercises 17 - 19

Use the precise definition of limit to prove the given one-sided limits.

17. \(\displaystyle \lim_{x →5^ −}\sqrt{5 −x}=0\)

Answer

Let \( δ= ε^2\). If \(- ε^2 < x - 5 < 0,\) we can multiply through by \(-1\) to get \(0 <5-x < ε^2.\)
Then \(\left|\sqrt{5 −x} - 0\right|=\sqrt{5 −x} < \sqrt{ ε^2} = ε\).

18. \(\displaystyle \lim_{x →0^+}f(x)= −2\), where \(f(x)=\begin{cases}8x −3, & \text{if }x <0\\4x −2, & \text{if }x ≥0\end{cases}\).
19. \(\displaystyle \lim_{x →1^ −}f(x)=3\), where \(f(x)=\begin{cases}5x −2, & \text{if }x <1\\7x −1, & \text{if }x ≥1\end{cases}\).

Answer

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

Exercises 20 - 22

Use the precise definition of limit to prove the given infinite limits.

20. \(\displaystyle \lim_{x →0}\frac{1}{x^2}= ∞\)
21. \(\displaystyle \lim_{x → −1}\frac{3}{(x+1)^2}= ∞\)

Answer

Let \( δ=\sqrt{\frac{3}{N}}\). If \(0 <|x+1| <\sqrt{\frac{3}{N}}\), then \(f(x)=\frac{3}{(x+1)^2} >N\).

22. \(\displaystyle \lim_{x →2} −\frac{1}{(x −2)^2}= − ∞\)

Exercises 23 - 28

22. 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 \( δ\), \( ε\), \(a\), and \(L\)?

Answer

\(0.033 \text{ cm}, \, ε=8,\, δ=0.33,\,a=12,\,L=144\)

24. Use the precise definition of limit to prove that the following limit does not exist: \(\displaystyle \lim_{x →1}\frac{|x −1|}{x −1}.\)
25. Using precise definitions of limits, prove that \(\displaystyle \lim_{x →0}f(x)\) does not exist, given that \(f(x)\) is the ceiling function. (Hint: Try any \( δ <1\).)

Answer

Answers may very.

26. Using precise definitions of limits, prove that \(\displaystyle \lim_{x →0}f(x)\) 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 \(0 <d\), >
27. Using precise definitions of limits, determine \(\displaystyle \lim_{x →0}f(x)\) 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

\(0\)

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

Exercises 29 - 31

Suppose that \(\displaystyle \lim_{x →a}f(x)=L\) and \(\displaystyle \lim_{x →a}g(x)=M\) both exist. Use the precise definition of limits to prove the following limit laws:

29. \(\displaystyle \lim_{x →a}(f(x) −g(x))=L −M\)

Answer

\(f(x) −g(x)=f(x)+( −1)g(x)\)

30. \(\displaystyle \lim_{x →a}[cf(x)]=cL\) for any real constant \(c\) (Hint: Consider two cases: \(c=0\) and \(c ≠0\).)
31. \(\displaystyle \lim_{x →a}[f(x)g(x)]=LM\). (Hint: \(|f(x)g(x) −LM|= |f(x)g(x) −f(x)M +f(x)M −LM| ≤|f(x)||g(x) −M| +|M||f(x) −L|.)\)

Answer

Answers may vary.