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Lecture Notes Limits at In
nity - Part 1 page 1
Sample Problems
1. Compute each of the following limits.
a) lim 3x4 c) lim ( 2x5) e) lim 2x6 g) lim 4x3
x!1 x!1 x!1 3 x!1
4 5 2 6 3
b) lim 3x d) lim ( 2x ) f) lim x h) lim 4x
x! 1 x! 1 x! 1 3 x! 1
2. Compute each of the following limits.
a) lim 1 d) lim 5 7+ 8 g) lim 5x3 2x+4
x!1x x! 1 2x3 x x!1 x2
b) lim 1 e) lim 2x3 +1 5 + 12 h) lim 5x3 2x+4
x! 1x x!1 x x4 x!1 x3
5 3x 2 5x3 2x+4
c) lim 3 f) lim i) lim 4
x!12x x! 1 x x!1 x
3. Compute each of the following limits.
a) lim ( 2x5 8x4 +7x3 10) c) lim ( 2x5 +8x6)
x! 1 x! 1
b) lim ( 2x5 8x4 +7x3 10) d) lim ( 2x5 +8x6)
x!1 x!1
4. Compute each of the following limits.
x+x2 6 x2 +9 x3 9x+1
a) lim 2 3 b) lim 2 c) lim 2
x! 16x+5x +2x x!15x+2x 3 x! 13x 2x 15
Practice Problems
1. Compute each of the following limits.
a) lim 3x15 c) lim 1x8 e) lim 4x9 g) lim ( 7x10)
x!1 8 x!13 x!1 x!1
b) lim 3x15 d) lim 1x8 f) lim 4x9 h) lim ( 7x10)
x! 1 8 x! 13 x! 1 x! 1
c
copyright Hidegkuti, Powell, 2009 Last revised: May 21, 2011
Lecture Notes Limits at In
nity - Part 1 page 2
2. Compute each of the following limits.
a) lim 3 g) lim 5x 2 m) lim 3x5+2x 5
x!1x5 x!1 x+3 x!1 x2
b) lim 3 h) lim 5x 2 n) lim 3x5+2x 5
x! 1x5 x! 1 x+3 x! 1 x2
c) lim 1 2 + 5 i) lim 5x 3 o) lim 4x8 +x3 x+7
x!1 x 3x4 x!1 x x!1 x4
d) lim 1 2 + 5 j) lim 5x 3 p) lim 4x8+x3 x+7
x! 1 x 3x4 x! 1 x x! 1 x4
e) lim 3+ 5 7 k) lim 1 3x
x!1 x3 6x x!1 2x
f) lim 3+ 5 7 l) lim 1 3x
x! 1 x3 6x x! 1 2x
3. Compute each of the following limits.
5 3 5 1 6 4 3 1
a) lim ( 7x +x ) c) lim 120x x e) lim 8x 3x x+2
x! 1 x! 1 4 x! 1 5
5 3 5 1 6 4 3 1
b) lim ( 7x +x ) d) lim 120x x f) lim 8x 3x x+2
x!1 x!1 4 x! 1 5
4. The graph of a polynomial function is shown on the picture below. What can we state about this
polynomial based on its end-behavior?
yy
xx
5. Compute each of the following limits.
a) lim 1 d) lim 3 2 + 11 g) lim 3x2 1
x! 1x x!1 x x4 x! 15x2 3x+2
5 2x2 +3x+1 20x 2x2 42
b) lim 3 e) lim 2 h) lim 3 2
x! 12x x!13x 5x+2 x! 15x 20x 105x
5 3x3+2x+1
c) lim 2 3 f) lim
x! 1 x x! 1 5x 3
c
copyright Hidegkuti, Powell, 2009 Last revised: May 21, 2011
Lecture Notes Limits at In
nity - Part 1 page 3
Sample Problems - Answers
1. a) 1 b) 1 c) 1 d) 1 e) 1 f) 1 g) 1 h) 1
2. a) 0 b) 0 c) 0 d) 7 e) 1 f) 3 g) 1 h) 5 i) 0
3. a) 1 b) 1 c) 1 d) 1
4. a) 0 b) 1 c) 1
2
Practice Problems - Answers
1. a) 1 b) 1 c) 1 d) 1 e) 1 f) 1 g) 1 h) 1
2. a) 0 b) 0 c) 1 d) 1 e) 3 f) 3 g) 1 h) 1 i) 5 j) 5 k) 3 l) 3
2 2
m) 1 n) 1 o) 1 p) 1
3. a) 1 b) 1 c) 1 d) 1 e) 1 f) 1
4. Since lim f (x) = 1 and lim f (x) = 1, the polynomial is of odd degree and has a positive
x! 1 x!1
leading coe¢ cient.
5. a) 0 b) 0 c) 2 d) 3 e) 2 f) 1 g) 3 h) 0
3 5
c
copyright Hidegkuti, Powell, 2009 Last revised: May 21, 2011
Lecture Notes Limits at In
nity - Part 1 page 4
Sample Problems - Solutions
1. Compute each of the following limits.
a) lim 3x4
x!1
Solution: Since the limit we are asked for is as x approaches in
nity, we should think of x as a very
large positive number. Then 3x4 is very large, and also positive because it is the product of
ve
positive numbers.
3x4 = 3 x x x x
positive positive positive positive positive
So the answer is 1. We state the answer: lim 3x4 = 1.
x!1
b) lim 3x4
x! 1
Solution: Since the limit we are asked for is as x approaches negative in
nity, we should think of x
as a very large negative number. Then 3x4 is very large, and also positive because it is the product
of one positive and four negative numbers.
3x4 = 3 x x x x
positive negative negative negative negative
So the answer is 1. We state the answer: lim 3x4 = 1
x! 1
c) lim ( 2x5)
x!1
Solution: Since the limit we are asked for is as x approaches in
nity, we should think of x as a very
large positive number. Then 2x5 is very large, and also negative because it is the product of one
negative and
ve positive numbers.
2x5 = 2 x x x x x
negative positive positive positive positive positive
So the answer is 1. We state the answer: lim ( 2x5) = 1
x!1
d) lim ( 2x5)
x! 1
Solution: Since the limit we are asked for is as x approaches negative in
nity, we should think of x
as a very large negative number. Then 2x5 is very large, and also positive because it is the product
of six negative numbers.
2x5 = 2 x x x x x
negative negative negative negative negative negative
So the answer is 1. We state the answer: lim ( 2x5) = 1
x! 1
c
copyright Hidegkuti, Powell, 2009 Last revised: May 21, 2011
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