Limits
1
Mathematics 53 2nd Semester, A.Y. 2014-2015 Exercises 5 - IVT, Squeeze Theorem, Limits IV Q3, R3, W8, X8 I. Use the Intermediate Value Theorem for the following. Before applying the theorem, verify if the assumptions hold. 1. Show that f (x)= x 3 - 4x + 1 has at least one solution in the interval [1, 2]. 2. Show that x 3 + x 2 - 2x = 1 has at least one solution in the interval [-1, 1]. 3. Show that f (x)= tan 2 x - 1 x - π has at least one solution in [0, π 3 ]. 4. Show that g(x)=3 x - 5x has a zero between x = 1 and x = 3. 5. Show that the function f (x)= x 4 - 5x 2 +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. lim x→-∞ 2x 2 + x sin x 2 - x 2 2. lim x→+∞ 2x + x cos(3x 2 ) 5x 2 - 2x +1 3. lim x→+∞ x 3 - cos √ x 1+2x 3 4. lim x→+∞ p sin(x 2 )+1 x 5 5. lim x→0 + √ x sin 1 2x 6. lim x→2 - √ 4 - x 2 sin 1 2 - x 7. Given |2g(x) - 5|≤ x - 4 for all x> 4, find lim x→4 + g(x). 8. Given |3f (x) - 5|≤ cos x, find lim x→ π 2 f (x). 9. lim x→-∞ 3 [[x - 2]] + 5 x (Hint: x - 1 ≤ [[x]] ≤ x) 10. lim x→+∞ 2x + x 2 tan -1 x x 4 +1 (Hint: - π 2 < tan -1 x< π 2 ) III. Evaluate the following limits involving trigonometric functions. 1. lim x→0 tan 2 3x x 3 +2x 2 2. lim x→0 x tan 5x sin 2 2x 3. lim x→0 1 - cos 2 3x 9x tan 5x 4. lim x→0 csc x - cot x 1 - csc x 5. lim x→0 sin 2 4x x 3 - 3x 2 6. lim x→0 1 - cos 5x cos 7x - 1 7. lim x→-2 sin(x 2 +3x + 2) x +2 8. lim x→3 x 2 - 7x + 12 sin(2x - 6) 9. lim x→ π 4 cot x - 1 x - π 4 (Hint: Let y = x - π 4 . Note that y → 0 as x → π 4 .) 10. lim x→0 c tan ax sin bx where a, b, c 6=0 Exercises from sample exams, books, and the internet rperez
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Exercises (Squeeze Theorem)
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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
