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2.3E: Limit Laws and Techniques for Computing Limits EXERCISES

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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).

83) limx0(4x22x+3)

Answer:

Use constant multiple law and difference law:

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

84) limx1x3+3x2+547x

85) limx2x26x+3

Answer:

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

86) limx1(9x+1)2

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

87) limx7x2

Answer:
49

88) limx2(4x21)

89) limx011+sinx

Answer:
1

90) limx2e2xx2

91) limx127xx+6

Answer:
57

92) limx3lne3x


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

93) limx4x216x4

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

94) limx2x2x22x

95) limx63x182x12

Answer:
limx63x182x12=18181212=00

then, limx63x182x12=limx63(x6)2(x6)=32

96) limh0(1+h)21h

97) limt9t9t3

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

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

99) limθπsinθtanθ

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

100) limx1x31x21

101) limx1/22x2+3x22x1

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

102) 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.

103) limx22x2+7x4x2+x2

Answer:
−∞

104) limx2+2x2+7x4x2+x2

105) limx12x2+7x4x2+x2

Answer:
−∞

106) 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

107) limx62f(x)g(x)

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

108) limx6g(x)1f(x)

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

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

110) limx6(h(x))32

111) limx6g(x)f(x)

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

112) limx6xh(x)

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

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

114) limx6(f(x)g(x)h(x))


[T] In the following exercises, use the definition of the piecewise-defined function to evaluate the given limits (you may want to draw the graph).

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

  1. a. limx3f(x)
  2. b. limx3+f(x)
  3. c. limx3f(x)
Answer:

CNX_Calc_Figure_02_03_202.jpeg

a. 9; b. 7; c. DNE

.

116) g(x)={x31x01x>0

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

117) h(x)={x22x+1x<23xx2

  1. a. limx2h(x)
  2. b. limx2+h(x)
  3. c. limx2h(x)
Answer:
a. 1; b. 1; c. 1

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

CNX_Calc_Figure_02_03_201.jpeg

118) limx3+(f(x)+g(x))

119) limx3(f(x)3g(x))

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

120) limx0f(x)g(x)3

121) limx52+g(x)f(x)

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

122) limx1(f(x))2

123) limx1f(x)g(x)

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

124) limx7(xg(x))

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

Answer:
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.

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

127) [T] limθ0θ2cos(1θ)

Answer:

The limit is zero.

CNX_Calc_Figure_02_03_206.jpeg

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

129) [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: 8.988×109N⋅m2/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?

Answer:

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.

130) [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

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

Answer:
5

214) limx03x22x+4

215) limx3x32x213x2

Answer:
8/7

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

217) limx5x2+25x+5

Answer:
DNE

218) limx23x22x8x24

219) limx1x21x31

Answer:
2/3

220) limx1x21x1

221) \\displaystyle limx44xx2

Answer:
−4

222) limx41x2


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

223) limx0x2cos(2πx)=0

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

224) limx0x3sin(πx)=0


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

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2.3: Limit Laws & Techniques for Computing Limits
2.4: Infinite Limits