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2.5E: Exercises for Section 2.5

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In exercises 1 - 4, write the appropriate εδ definition for each of the given statements.

1) limxaf(x)=N

2) limtbg(t)=M

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

3) limxch(x)=L

4) limxaφ(x)=A

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

The following graph of the function f satisfies limx2f(x)=2. In the following exercises, determine a value of δ>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<|x2|<δ, then |f(x)2|<1.

6) If 0<|x2|<δ, then |f(x)2|<0.5.

Answer
δ0.25

The following graph of the function f satisfies limx3f(x)=1. In the following exercises, determine a value of δ>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<|x3|<δ, then |f(x)+1|<1.

8) If 0<|x3|<δ, then |f(x)+1|<2.

Answer
δ2

The following graph of the function f satisfies limx3f(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.

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) ε=1.5

10) ε=3

Answer
δ1

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

11) |sin(2x)12|<0.1, whenever |xπ12|<δ

12) |x42|<0.1, whenever |x8|<δ

Answer
δ<0.3900

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

13) limx2(5x+8)=18

14) limx3x29x3=6

Answer
Let δ=ε. If 0<|x3|<ε, then |x29x36|=|(x+3)(x3)x36|=|x+36|=|x3|<ε.

15) limx22x23x2x2=5

16) limx0x4=0

Answer
Let δ=4ε. If 0<|x|<4ε, then |x40|=x4<ε.

17) limx2(x2+2x)=8

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

18) limx55x=0

Answer
Let δ=ε2. If ε2<x5<0, we can multiply through by 1 to get 0<5x<ε2.
Then |5x0|=5x<ε2=ε.

19) limx0+f(x)=2, where f(x)={8x3,if x<04x2,if x0.

20) limx1f(x)=3, where f(x)={5x2,if x<17x1,if x1.

Answer
Let δ=ε/5. If ε/5<x1<0, we can multiply through by 1 to get 0<1x<ε/5.
Then |f(x)3|=|5x23|=|5x5|=5(1x), since x<1 here.
And 5(1x)<5(ε/5)=ε.

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

21) limx01x2=

22) limx13(x+1)2=

Answer
Let δ=3N. If 0<|x+1|<3N, then f(x)=3(x+1)2>N.

23) limx21(x2)2=

24) An engineer is using a machine to cut a flat square of Aerogel of area 144cm2. If there is a maximum error tolerance in the area of 8cm2, 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 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: limx1|x1|x1.

26) Using precise definitions of limits, prove that limx0f(x) does not exist, given that f(x) is the ceiling function. (Hint: Try any δ<1.)

Answer
Answers may very.

27) Using precise definitions of limits, prove that limx0f(x) does not exist: f(x)={1,if x is rational0,if x is irrational. (Hint: Think about how you can always choose a rational number \(0 <d\), >

28) Using precise definitions of limits, determine limx0f(x) for f(x)={x,if x is rational0,if x is irrational. (Hint: Break into two cases, x rational and x irrational.)

Answer
0

29) Using the function from the previous exercise, use the precise definition of limits to show that limxaf(x) does not exist for a0

For exercises 30 - 32, suppose that limxaf(x)=L and limxag(x)=M both exist. Use the precise definition of limits to prove the following limit laws:

30) limxa(f(x)g(x))=LM

Answer
f(x)g(x)=f(x)+(1)g(x)

31) limxa[cf(x)]=cL for any real constant c (Hint: Consider two cases: c=0 and c0.)

32) limxa[f(x)g(x)]=LM. (Hint: |f(x)g(x)LM|=|f(x)g(x)f(x)M+f(x)MLM||f(x)||g(x)M|+|M||f(x)L|.)

Answer
Answers may vary.

This page titled 2.5E: Exercises for Section 2.5 is shared under a not declared license and was authored, remixed, and/or curated by Zoya Kravets.

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