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NEW 2.3E: Limit Laws

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2.3: The Limit Laws

In the following exercises, use the limit laws to evaluate each limit. Justify each step by indicating the appropriate limit law(s).

1) limx0(4x22x+3)

Solution: Use constant multiple law and difference law:

limx0(4x22x+3)=4limx0x22limx0x+limx03=3

2) limx1x3+3x2+547x

3) limx2x26x+3

Solution: Use root law: limx2x26x+3=limx2(x26x+3)=19

4) limx1(9x+1)2

In the following exercises, use direct substitution to evaluate each limit.

5) limx7x2)

Solution: 49

6) limx2(4x21)

7) limx011+sinx

Solution: 1

8) limx2e2xx2

9) limx127xx+6

Solution: 57

10) limx3lne3x

In the following exercises, use direct substitution to show that each limit leads to the indeterminate form 0/0. Then, evaluate the limit.

11) limx4x216x4

Solution:limx4x216x4=161644=00;then,limx4x216x4=limx4(x+4)(x4)x4=8

12) limx2x2x22x

13) limx63x182x12

Solution: limx63x182x12=18181212=00;then,limx63x182x12=limx63(x6)2(x6)=32

14) limh0(1+h)21h

15) limt9t9t3

Solution: limx9t9t3=9933=00;then,limt9t9t3=limt9t9t3t+3t+3=limt9(t+3)=6

16) limh01a+h1ah, where a is a real-valued constant

17) limθπsinθtanθ

Solution: limθπsinθtanθ=sinπtanπ=00;then,limθπsinθtanθ=limθπsinθsinθcosθ=limθπcosθ=1

18) limx1x31x21

19) limx1/22x2+3x22x1

Solution: limx1/22x2+3x22x1=12+32211=00;then,limx1/22x2+3x22x1=limx1/2frac(2x1)(x+2)2x1=52

20) limx3x+41x+3

In the following exercises, use direct substitution to obtain an undefined expression. Then, use the method of Example to simplify the function to help determine the limit.

21) limx22x2+7x4x2+x2

Solution: −∞

22) limx2+2x2+7x4x2+x2

23) limx12x2+7x4x2+x2

Solution: −∞

24) limx1+2x2+7x4x2+x2

In the following exercises, assume that limx6f(x)=4,limx6g(x)=9, and limx6h(x)=6. Use these three facts and the limit laws to evaluate each limit

25) limx62f(x)g(x)

Solution: limx62f(x)g(x)=2limx6f(x)limx6g(x)=72

26) limx6g(x)1f(x)

27) limx6(f(x)+13g(x))

Solution: limx6(f(x)+13g(x))=limx6f(x)+13limx6g(x)=7\

28) limx6(h(x))32

29) limx6g(x)f(x)

Solution: limx6g(x)f(x)=limx6g(x)limx6f(x)=5

30) limx6xh(x)

31) limx6[(x+1)f(x)]

Solution: limx6[(x+1)f(x)]=(limx6(x+1))(limx6f(x))=28

32) limx6(f(x)g(x)h(x))

[T] In the following exercises, use a calculator to draw the graph of each piecewise-defined function and study the graph to evaluate the given limits.

33) f(x)={x2x3,x+4x>3

  1. a. limx3f(x)
  2. b. limx3+f(x)

Solution:

CNX_Calc_Figure_02_03_202.jpeg

a. 9; b. 7

34) g(x)={x31x01x>0

  1. a. limx0g(x)
  2. b. limx0+g(x)

35) h(x)={x22x+1x<2x23xx2

  1. a. limx2h(x)
  2. b. limx2+h(x)

In the following exercises, use the following graphs and the limit laws to evaluate each limit.

CNX_Calc_Figure_02_03_201.jpeg

36) limx3+(f(x)+g(x))

37) limx3(f(x)3g(x))

Solution: limx3(f(x)3g(x))=limx3f(x)3limx3g(x)=0+6=6

38) limx0f(x)g(x)3

39) limx52+g(x)f(x)

Solution: limx52+g(x)f(x)=2+(limx5g(x))limx5f(x)=2+02=1

40) limx1(f(x))2

41) limx1f(x)g(x)

Solution: limx13f(x)g(x)=3limx1f(x)limx1g(x)=32+5=37

42) limx7(xg(x))

43) limx9[xf(x)+2g(x)]

Solution: limx9(xf(x)+2g(x))=(limx9x)(limx9f(x))+2limx9(g(x))=(9)(6)+2(4)=46

For the following problems, evaluate the limit using the squeeze theorem. Use a calculator to graph the functions f(x),g(x), and h(x) when possible.

44) [T] True or False? If 2x1g(x)x22x+3, then limx2g(x)=0.

45) [T] \(lim_{θ→0}θ^2cos(\frac{1}{θ})

Solution: The limit is zero.

CNX_Calc_Figure_02_03_206.jpeg

46) limx0f(x), where f(x)={0xrationalx2xirrrational

47) [T] In physics, the magnitude of an electric field generated by a point charge at a distance r in vacuum is governed by Coulomb’s law: E(r)=q4πε02r, where E represents the magnitude of the electric field, q is the charge of the particle, r is the distance between the particle and where the strength of the field is measured, and \frac{1}{4πε_0} is Coulomb’s constant: 8.988×109Nm2/C2.

a. Use a graphing calculator to graph E(r) given that the charge of the particle is q=1010.

b. Evaluate limr0+E(r). What is the physical meaning of this quantity? Is it physically relevant? Why are you evaluating from the right?

Solution: a

CNX_Calc_Figure_02_03_207.jpeg

b. ∞. The magnitude of the electric field as you approach the particle q becomes infinite. It does not make physical sense to evaluate negative distance.

48) [T] The density of an object is given by its mass divided by its volume: ρ=m/V.

a. Use a calculator to plot the volume as a function of density (V=m/ρ), assuming you are examining something of mass 8 kg (m=8).

b. Evaluate limx0+V(ρ) and explain the physical meaning.

Chapter Review Exercises

True or False. In the following exercises, justify your answer with a proof or a counterexample.

212) Using the graph, find each limit or explain why the limit does not exist.

a. limx1f(x)

b. limx1f(x)

c. limx0+f(x)

d. limx2f(x)

CNX_Calc_Figure_02_05_207.jpeg


In the following exercises, evaluate the limit algebraically or explain why the limit does not exist.

213) limx22x23x2x2

Solution: 5

214) limx03x22x+4

215) limx3x32x213x2

Solution: 8/7

216) limxπ/2cotxcosx This is covered in section 2.4

217) limx5x2+25x+5

Solution:DNE

218) limx23x22x8x24

219) limx1x21x31

Solution: 2/3

220) limx1x21x1

221) limx44xx2

Solution: −4

222) limx41x2


In the following exercises, use the squeeze theorem to prove the limit.

223) limx0x2cos(2πx)=0

Solution: Since 1cos(2πx)1, then x2x2cos(2πx)x2. Since limx0x2=0=limx0x2, it follows that limx0x2cos(2πx)=0.

224) limx0x3sin(πx)=0


NEW 2.3E: Limit Laws is shared under a CC BY-NC-SA license and was authored, remixed, and/or curated by LibreTexts.

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