# 2.6E: Exercises for Section 2.5

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In exercises 1 - 4, write the appropriate $$ε − δ$$ definition for each of the given statements.

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

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

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

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

4) $$\displaystyle \lim_{x →a} φ(x)=A$$

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

The following graph of the function $$f$$ satisfies $$\displaystyle \lim_{x →2}f(x)=2$$. In the following exercises, determine a value of $$δ >0$$ that satisfies each statement. 5) If $$0 <|x −2| < δ$$, then $$|f(x) −2| <1$$.

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

$$δ ≤0.25$$

The following graph of the function $$f$$ satisfies $$\displaystyle \lim_{x →3}f(x)= −1$$. In the following exercises, determine a value of $$δ >0$$ that satisfies each statement. 7) If $$0 <|x −3| < δ$$, then $$|f(x)+1| <1$$.

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

$$δ ≤2$$

The following graph of the function $$f$$ 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. 9) $$ε=1.5$$

10) $$ε=3$$

$$δ ≤1$$

[T] In exercises 11 - 12, use a graphing calculator to find a number $$δ$$ such that the statements hold true.

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

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

$$δ <0.3900$$

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

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

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

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

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

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

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

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

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

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

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

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

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

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 $$5(1-x) < 5( ε/5) = ε$$.

In exercises 21 - 23, use the precise definition of limit to prove the given infinite limits.

21) $$\displaystyle \lim_{x →0}\frac{1}{x^2}= ∞$$

22) $$\displaystyle \lim_{x → −1}\frac{3}{(x+1)^2}= ∞$$

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

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

24) 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 $$L$$?

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

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

26) 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 $$δ <1$$.)

27) 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 $$0 <d$$, >

28) 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.)

$$0$$

29) 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 $$a ≠0$$

For exercises 30 - 32, 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:

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

$$f(x) −g(x)=f(x)+( −1)g(x)$$
31) $$\displaystyle \lim_{x →a}[cf(x)]=cL$$ for any real constant $$c$$ (Hint: Consider two cases: $$c=0$$ and $$c ≠0$$.)
32) $$\displaystyle \lim_{x →a}[f(x)g(x)]=LM$$. (Hint: $$|f(x)g(x) −LM|= |f(x)g(x) −f(x)M +f(x)M −LM| ≤|f(x)||g(x) −M| +|M||f(x) −L|.)$$