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Horizontal and Vertical Asymptotes Worksheet

Horizontal and Vertical Asymptotes Worksheet
  • Page 1
 1.  
Which of the following is the horizontal asymptote of the curve y = 5x?
a.
x + 5 = 0
b.
x = 0
c.
x = 5
d.
y = 0


Solution:

y = f (x) = 5x

If either limx+ ∞ f (x) = l or limx- ∞ f (x) = l or both, then the line y = l is the horizontal asymptote of the curve f (x) .
[Definition.]

limx± ∞ f (x) = limx± ∞ 5x
[Substitute f (x) = 5x .]

= 0
[Evaluate.]

So, the line y = 0 or the x-axis is the horizontal asymptote of the curve y = 5x


Correct answer : (4)
 2.  
Which of the following is the horizontal asymptote of the curve y = 7x2?
a.
x = 7
b.
y = 0
c.
x = 0
d.
y = 7


Solution:

y = f (x) = 7x2

If either limx+ ∞ f (x) = l or limx- ∞ f (x) = l or both, then the line y = l is the horizontal asymptote of the curve f (x) .
[Definition.]

limx± ∞ f (x) = l
[For horizontal asymptotes.]

limx± ∞ 7x2 = 0
[Evaluate.]

So, the line y = 0 is the horizontal asymptote of the curve f (x) = 7x2.


Correct answer : (2)
 3.  
Which of the following is the horizontal asymptote of the curve f (x) = 6x + 3x + 4?
a.
y = 6
b.
x = 6
c.
x = - 6
d.
y = - 6


Solution:

y = 6x + 3x + 4

If either limx f (x) = l or limx-∞ f (x) = l or both, then the line y = l is the horizontal asymptote of the curve f (x) .
[Definition.]

limx±∞ f (x)
[For horizontal asymptotes.]

= limx±∞ (6x + 3x + 4)
[Substitute f (x) = 6x + 3x + 4.]

= limx±∞ 6+3x1+4x
[Divide both the numerator, denominator by x.]

= 6 + 01 + 0 = 6

So, the line y = 6 is the horizontal asymptote of the curve f (x) = 6x + 3x + 4.


Correct answer : (1)
 4.  
Which of the following is the horizontal asymptote of the curve f (x) = x - 35x - 4?
a.
y = - 1 5
b.
y = 1 5
c.
y + 5 = 0
d.
y = 0


Solution:

y = f (x) = x - 35x - 4

If either limx+ ∞ f (x) = l or limx- ∞ f (x) = l or both, then the line y = l is the horizontal asymptote of the curve f (x).
[Definition.]

limx±∞ f (x)
[For horizontal asymptotes.]

= limx±∞ x - 35x - 4
[Substitute f (x) = x - 35x - 4.]

= limx±∞ (1 -3x)(5 -4x)
[Divide both the numerator, denominator by x.]

= 1 - 05 - 0 = 1 / 5

So, the line y = 1 / 5 is the horizontal asymptote of the curve f (x) = x - 35x - 4


Correct answer : (2)
 5.  
Which of the following is the horizontal asymptote of the curve f (x) = 6x2 + 7x2 + 9?
a.
y = - 6
b.
x = - 6
c.
y = 6
d.
x = 6


Solution:

y = f (x) = 6x2 + 7x2 + 9

If either limx+ ∞ f (x) = l or limx- ∞ f (x) = l or both, then the line y = l is the horizontal asymptote of the curve f (x).
[Definition.]

limx±∞ f (x)
[For horizontal asymptotes.]

= limx±∞ 6x2 + 7x2 + 9
[Substitute f (x) = 6x2 + 7x2 + 9.]

= limx±∞(6+7x2)(1+9x2)
[Divide both the numerator, denominator by x2.]

= 6 + 01 + 0 = 6

So, the line y = 6 is the horizontal asymptote of the curve f (x) = 6x2 + 7x2 + 9


Correct answer : (3)
 6.  
Which of the following is the horizontal asymptote of the curve f (x) = 2e-x?
a.
y = 0
b.
y = 2
c.
x = 0
d.
y = - 2


Solution:

y = f (x) = 2e-x

If either limx+∞ f (x) = l or limx-∞ f (x) = l or both, then the line y = l is the horizontal asymptote of the curve f (x)
[Definition.]

limx+∞ f (x)
[For horizontal asymptotes.]

= limx+∞ 2e-x
[Substitute f (x) = 2e-x.]

= limx+∞ 2ex = 0

limx-∞ f (x)
[For horizontal asymptote.]

= limx-∞ 2e-x = ∞

So, the line y = 0 or the x- axis is the horizontal asymptote of the curve f (x).


Correct answer : (1)
 7.  
Which of the following is the horizontal asymptote of the curve f (x) = e5x?
a.
y = - 5
b.
y = 5
c.
y = 0
d.
x = 0


Solution:

y = f (x) = e5x

If either limx+ ∞ f (x) = l or limx- ∞ f (x) = l or both, then the line y = l is the horizontal asymptote of the curve f (x).

limx+ ∞ f (x)
[For horizontal asymptote.]

= limx+ ∞ e5x = ∞
[Substitute f (x) = e5x.]

limx- ∞ f (x)
[For horizontal asymptote.]

= limx- ∞ e5x = 0
[Substitute f (x) = e5x.]

So, the line y = 0 or the x-axis is the horizontal asymptote of the curve f (x).


Correct answer : (3)
 8.  
Which of the following is the horizontal asymptote of the curve f (x) = 8e- 4x2?
a.
x = 0
b.
y + 8 = 0
c.
y = 0
d.
y = 8


Solution:

y = f (x) = 8e- 4x2

If either limx f (x) = l or limx-∞ f (x) = l or both, then the line y = l is the horizontal asymptote of the curve f (x).
[Definition.]

limx±∞ f (x)
[For horizontal asymptotes.]

= limx±∞ 8e- 4x2
[Substitute f (x) = 8e- 4x2.]

= limx±∞ (8e4x2)

= 0

So, the line y = 0 or the x-axis is the horizontal asymptote of the curve f (x).


Correct answer : (3)
 9.  
Which of the following is the horizontal asymptote of the curve f (x) = 8x2 - 649x2 + 64?
a.
y = 8
b.
x = 8 9
c.
y = 8 9
d.
y = 9


Solution:

y = f (x) = 8x2 - 649x2 + 64

If either limx+∞ f (x) = l or limx-∞ f (x) = l or both, then the line y = l is the horizontal asymptote of the curve f (x).

limx±∞ f (x)
[For horizontal asymptotes.]

= limx±∞ 8x2 - 649x2 + 64
[Substitute f (x) from step1.]

= limx±∞(8 -64x2)(9 +64x2)
[Divide the both numerator, denominator by x2.]

= 8 - 09 + 0 = 8 / 9

So, the line y = 8 / 9 is the horizontal asymptote of the curve f (x).


Correct answer : (3)
 10.  
Which of the following is the horizontal asymptote of the curve f (x) = 7x211x2 + 14?
a.
y = 0
b.
y = 1 2
c.
y = 7 11
d.
y = 7 25


Solution:

y = f (x) = (7x211x2+14)

If either limx+ ∞ f (x) = l or limx- ∞ f (x) = l or both, then the line y = l is the horizontal asymptote of the curve f (x).

limx± ∞ f (x)
[For horizontal asymptotes.]

= limx± ∞ (7x211x2+14)
[Substitute f (x) from step-1.]

= limx± ∞ 7(11+14x2)
[Divide both the numerator, denominator by x2.]

= 711+0 = 7 / 11

So, the line y = 7 / 11 is the horizontal asymptote of the curve f (x).


Correct answer : (3)

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