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2.5E: 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 $$0<r<d$$, but $$|f(r)−0|=1$$.)

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: $$|f(x)g(x)−LM|= |f(x)g(x) −f(x)M +f(x)M −LM| ≤|f(x)||g(x) −M| +|M||f(x)−L|.)$$