To get the best deal on Tutoring, call 1-855-666-7440 (Toll Free)

Limit Worksheets

Limit Worksheets
  • Page 1
 1.  
Evaluate limx7 8x - 56x - 7.
a.
10
b.
8
c.
9
d.
7


Solution:

limx7 8x -56x - 7

= limx78(x - 7)x - 7

= limx7 8

= 8.


Correct answer : (2)
 2.  
Find the value of limx7 x - 7x2 - 14x + 49.
a.
- 1 5
b.
- 1 7
c.
does not exist


Solution:

limx7 x - 7x2 - 14x + 49

= limx7 x - 7(x - 7)2

= limx7 1x - 7

= ± ∞

So , the limit does not exist.


Correct answer : (3)
 3.  
What is the value of limx5 5(x - 5)x2 - 6x + 5?
a.
- 5
b.
54
c.
5 6


Solution:

limx5 5(x - 5)x2 - 6x + 5

= limx5 5(x - 5)(x - 1)(x - 5)
[Factor x2 - 6 x + 5.]

= limx5 5x - 1

= 55 - 1

= 5 / 4


Correct answer : (3)
 4.  
Find the value of limx02x - 3x2x.
a.
2
b.
3
c.
-1


Solution:

limx02x - 3x2x

= limx0x(2 - 3x)x
[Factor 2x - 3x2.]

= limx0 (2 - 3x)

= 2 - 3(0)

= 2.


Correct answer : (1)
 5.  
Evaluate limx3 x2 - 9x - 3.
a.
6
b.
3
c.


Solution:

limx3 x2 - 9x - 3

= limx3 (x - 3)(x + 3)x - 3
[Factor x2 - 9.]

= limx3 (x + 3)

= (3 + 3) = 6.


Correct answer : (1)
 6.  
Evaluate limx1x2 + 81 -82x - 1.
a.
182
b.
does not exist
c.


Solution:

limx1x2 + 81 -82x - 1

= limx1 (x2 + 81-82)(x2 + 81 +82)(x - 1)(x2 + 81 +82)
[Rationalize the numerator.]

= limx1 x2 - 1(x - 1)(x2 + 81 +82)

= limx1 (x +1)(x - 1)(x - 1)(x2 + 81 +82)
[Factor (x2 - 1).]

= limx1 x + 1(x2 + 81 +82)
[Cancel the common factor.]

= 2282 = 182.


Correct answer : (1)
 7.  
Find the equation of the tangent to the curve f (x) = x2 such that f (8) = 64.
a.
y - 16x - 64 = 0
b.
y = 16x + 64
c.
16x - y - 64 = 0
d.
None of the above


Solution:

f (x) = x2, f (8) = 64.

limh0f (8 + h) - f (8)h = limh0(8 + h )2 - 64h = limh0 64 + 16h +h2 - 64h

= limh016 + h =16.

So, the slope of the tangent line to the curve at the point (8, 64) = 16.

The equation of the tangent line to the curve at the point (8, 64) is y - 64 = 16 (x - 8) or y = 16x - 64.
[Use slope point form.]


Correct answer : (3)
 8.  
Find the equation of the tangent to the curve f (x) = 1 - 2x + x2 ; f (-4) = 25.
a.
y - 2x = 0
b.
y + x = 0
c.
y + 10 x = -15
d.
y - 10x = 0


Solution:

f (x) = 1 - 2x + x2, f (-4) = 25

f (-4 + h) - f (-4)h = 1 - 2(-4 + h) +(-4 + h)2 - 25h = h - 10

limh0 f (-4 + h) - f (-4)h = limh0 ( h - 10) = - 10

So, the slope of the tangent to the curve at the point (-4, 25) = - 10

The equation of the tangent line to the curve at the point (-4, 25) is y - 25 = - 10(x-(-4)) y + 10x = -15.
[Use slope point form.]


Correct answer : (3)
 9.  
Find the equation of the tangent to the curve f (x) = x, f (8) = 8.
a.
y = x + 82
b.
y = x28 + 82
c.
y = x - 1
d.
y = x - 12


Solution:

f (x) = x; f (8) = 8

f (8 + h) - f (8)h = 8 + h -8 h

= 8 + h -8h . 8 + h +88 + h +8

= hh(8 + h +8)

= 18 + h +8

limh0 f (1 + h) - f (1)h = limh0 18 + h +8 = 128

So , the slope of the tangent line to the curve at the point (8,8) = 128 .

The equation of the tangent line to the curve at the point (8,8) is y - 8 = 128(x - 8) or y = x28 + 82.
[Use slope point form.]


Correct answer : (2)
 10.  
Which of the following is the tangent to the curve f (x) = x5 + 4 such that f (0) = 4?
a.
y + x = 1
b.
y = 0
c.
y = x
d.
y = 4


Solution:

f (x) = x5 + 4; f (0) = 4

f (0 + h) - f (0)h = (h5 + 4) - (0 + 4)h = h4

limh0 f (0 + h) - f (0)h = limh0 h4 = 0

So, The slope of the tangent line to the curve at the poiunt (0,4) = 0.

The equation of the tangent line to the curve at the point (0,4) is y - 4 = 0(x - 0) y = 4
[Use slope point form.]


Correct answer : (4)

More Limit Worksheets
Limits of Polynomial and Rational Functions Worksheet Investigating Limits Using Tables and Graphs Worksheet
Limits and Asymptotes Worksheet Limits and Continuity Worksheet
Limits and Motion: the Area Problem Worksheet Limits at Infinity Worksheet
Limits of Exponential and Logarithmic Functions Worksheet Limits of Functions Involving Modulus and Greatest Integer Worksheet
Limits of Functions Worksheet Limits of Piece Wise Functions Worksheet
Limits of Trigonometric and Radical Functions Worksheet Power and Sum/difference Rules of Limits Worksheet
Product and Quotient Rules of Limits Worksheet Central Limit Theorem Worksheet
Continuity and One Sided Limits Worksheet Evaluate Limits Using Rules Worksheet
Evaluating Limits Analytically Worksheet Evaluating Limits Using Definition (epsilon - Delta) Worksheet
Infinite Limits Worksheet L'hopital's Rule Worksheet
The Squeeze Theorem Worksheet
*AP and SAT are registered trademarks of the College Board.