Mathematics 53 2nd Semester, A.Y. 2014-2015Exercises 5 - IVT, Squeeze Theorem, Limits IV Q3, R3, W8, X8

I. Use the Intermediate Value Theorem for the following. Before applying the theorem, verify ifthe assumptions hold.

1. Show that f(x) = x3 4x+ 1 has at least one solution in the interval [1, 2].2. Show that x3 + x2 2x = 1 has at least one solution in the interval [1, 1].

3. Show that f(x) =tan2 x 1x pi has at least one solution in [0,

pi3].

4. Show that g(x) = 3x 5x has a zero between x = 1 and x = 3.5. Show that the function f(x) = x4 5x2 + 2x + 1 has at least two zeros on [0, 3] without

identifying the zeros.

II. Use the Squeeze Theorem to evaluate the following limits.

1. limx

2x2 + x sinx

2 x2

2. limx+

2x+ x cos(3x2)

5x2 2x+ 1

3. limx+

x3 cosx1 + 2x3

4. limx+

sin(x2) + 1

x5

5. limx0+

x sin

(1

2x

)6. lim

x2

4 x2 sin(

1

2 x)

7. Given |2g(x) 5| x 4 for all x > 4,find lim

x4+g(x).

8. Given |3f(x) 5| cosx, find limxpi

2

f(x).

9. limx

3 [[x 2]] + 5x

(Hint: x 1 [[x]] x)

10. limx+

2x+ x2 tan1 xx4 + 1

(Hint: pi2< tan1 x < pi

2)

III. Evaluate the following limits involving trigonometric functions.

1. limx0

tan2 3x

x3 + 2x2

2. limx0

x tan 5x

sin2 2x

3. limx0

1 cos2 3x9x tan 5x

4. limx0

cscx cotx1 cscx

5. limx0

sin2 4x

x3 3x2

6. limx0

1 cos 5xcos 7x 1

7. limx2

sin(x2 + 3x+ 2)

x+ 2

8. limx3

x2 7x+ 12sin(2x 6)

9. limxpi

4

cotx 1x pi

4

(Hint: Let y = x pi4.

Note that y 0as x pi

4.)

10. limx0

c tan ax

sin bx

where a, b, c 6= 0

Exercises from sample exams, books, and the internet rperez