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Mathematics LibreTexts

2.3E: Exercises for Section 2.3

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In exercises 1 - 4, use the limit laws to evaluate each limit. Justify each step by indicating the appropriate limit law(s).

1) limx0(4x22x+3)

Answer

Use constant multiple law and difference law:

limx0(4x22x+3)=4limx0x22limx0x+limx03=0+0+3=3

2) limx1x3+3x2+547x

3) limx2x26x+3

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

4) limx1(9x+1)2

In exercises 5 - 10, use direct substitution to evaluate the limit of each continuous function.

5) limx7x2

Answer
limx7x2=49

6) limx2(4x21)

7) limx011+sinx

Answer
limx011+sinx=1

8) limx2e2xx2

9) limx127xx+6

Answer
limx127xx+6=57

10) limx3lne3x

In exercises 11 - 20, use direct substitution to show that each limit leads to the indeterminate form 0/0. Then, evaluate the limit analytically.

11) limx4x216x4

Answer
When x=4,x216x4=161644=00;

then, limx4x216x4=limx4(x+4)(x4)x4=limx4(x+4)=4+4=8

12) limx2x2x22x

13) limx63x182x12

Answer
When x=6,3x182x12=18181212=00;

then, limx63x182x12=limx63(x6)2(x6)=limx632=32

14) limh0(1+h)21h

15) limt9t9t3

Answer
When t=9,t9t3=9933=00;

then, limt9t9t3=limt9t9t3t+3t+3=limt9(t9)(t+3)t9=limt9(t+3)=9+3=6

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

17) limθπsinθtanθ

Answer
When θ=π,sinθtanθ=sinπtanπ=00;

then, limθπsinθtanθ=limθπsinθsinθcosθ=limθπcosθ=cosπ=1

18) limx1x31x21

19) limx1/22x2+3x22x1

Answer
When x=1/2,2x2+3x22x1=12+32211=00;

then, limx1/22x2+3x22x1=limx1/2(2x1)(x+2)2x1=limx1/2(x+2)=12+2=52

20) limx3x+41x+3

In exercises 21 - 24, use direct substitution to obtain an undefined expression. Then, use the method used in Example 9 of this section to simplify the function and determine the limit.

21) limx22x2+7x4x2+x2

Answer

22) limx2+2x2+7x4x2+x2

23) limx12x2+7x4x2+x2

Answer

24) limx1+2x2+7x4x2+x2

In exercises 25 - 32, 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)

Answer
limx62f(x)g(x)=2(limx6f(x))(limx6g(x))=2(4)(9)=72

26) limx6g(x)1f(x)

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

Answer
limx6(f(x)+13g(x))=limx6f(x)+13limx6g(x)=4+13(9)=7

28) limx6(h(x))32

29) limx6g(x)f(x)

Answer
limx6g(x)f(x)=limx6g(x)limx6f(x)=94=5

30) limx6xh(x)

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

Answer
limx6[(x+1)f(x)]=(limx6(x+1))(limx6f(x))=7(4)=28

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

[T] In exercises 33 - 35, use a calculator to draw the graph of each piecewise-defined function and study the graph to evaluate the given limits.

33) f(x)={x2,x3x+4,x>3

a. limx3f(x)

b. limx3+f(x)

Answer

The graph of a piecewise function with two segments. The first is the parabola x^2, which exists for x<=3. The vertex is at the origin, it opens upward, and there is a closed circle at the endpoint (3,9). The second segment is the line x+4, which is a linear function existing for x > 3. There is an open circle at (3, 7), and the slope is 1.

a. 9; b.7

34) g(x)={x31,x01,x>0

a. limx0g(x)

b. limx0+g(x)

35) h(x)={x22x+1,x<23x,x2

a. limx2h(x)

b. limx2+h(x)

In exercises 36 - 43, use the following graphs and the limit laws to evaluate each limit.

Two graphs of piecewise functions. The upper is f(x), which has two linear segments. The first is a line with negative slope existing for x < -3. It goes toward the point (-3,0) at x= -3. The next has increasing slope and goes to the point (-3,-2) at x=-3. It exists for x > -3. Other key points are (0, 1), (-5,2), (1,2), (-7, 4), and (-9,6). The lower piecewise function has a linear segment and a curved segment. The linear segment exists for x < -3 and has decreasing slope. It goes to (-3,-2) at x=-3. The curved segment appears to be the right half of a downward opening parabola. It goes to the vertex point (-3,2) at x=-3. It crosses the y axis a little below y=-2. Other key points are (0, -7/3), (-5,0), (1,-5), (-7, 2), and (-9, 4).

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

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

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

38) limx0f(x)g(x)3

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

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

40) limx1(f(x))2

41) limx13f(x)g(x)

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

42) limx7(xg(x))

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

Answer
limx9(xf(x)+2g(x))=(limx9x)(limx9f(x))+2limx9g(x)=(9)(6)+2(4)=46

For exercises 44 - 46, 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] limθ0θ2cos(1θ)

Answer

The limit is zero.

The graph of three functions over the domain [-1,1], colored red, green, and blue as follows: red: theta^2, green: theta^2 * cos (1/theta), and blue: - (theta^2). The red and blue functions open upwards and downwards respectively as parabolas with vertices at the origin. The green function is trapped between the two.

46) limx0f(x), where f(x)={0,x rationalx2,x irrrational

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)=q4πε0r2, 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 14πε0 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?

Answer

a.

A graph of a function with two curves. The first is in quadrant two and curves asymptotically to infinity along the y axis and to 0 along the x axis as x goes to negative infinity. The second is in quadrant one and curves asymptotically to infinity along the y axis and to 0 along the x axis as x goes to infinity.

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.


2.3E: Exercises for Section 2.3 is shared under a not declared license and was authored, remixed, and/or curated by LibreTexts.

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