Horizontal and Vertical Asymptotes Worksheet

Horizontal and Vertical Asymptotes Worksheet
• Page 1
1.
Which of the following is the horizontal asymptote of the curve $y$ = $\frac{5}{x}$?
 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

2.
Which of the following is the horizontal asymptote of the curve $y$ = $\frac{7}{{x}^{2}}$?
 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.

3.
Which of the following is the horizontal asymptote of the curve $f$ ($x$) = ?
 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.

4.
Which of the following is the horizontal asymptote of the curve $f$ ($x$) = ?
 a. $y$ = - $\frac{1}{5}$ b. $y$ = $\frac{1}{5}$ 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

5.
Which of the following is the horizontal asymptote of the curve $f$ ($x$) = ?
 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

6.
Which of the following is the horizontal asymptote of the curve $f$ ($x$) = 2$e$-$x$?
 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).

7.
Which of the following is the horizontal asymptote of the curve $f$ ($x$) = $e$5$x$?
 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).

8.
Which of the following is the horizontal asymptote of the curve $f$ ($x$) = 8$e$- 4$x$2?
 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).

9.
Which of the following is the horizontal asymptote of the curve $f$ ($x$) = ?
 a. $y$ = 8 b. $x$ = $\frac{8}{9}$ c. $y$ = $\frac{8}{9}$ 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).

10.
Which of the following is the horizontal asymptote of the curve $f$ ($x$) = ?
 a. $y$ = 0 b. $y$ = $\frac{1}{2}$ c. $y$ = $\frac{7}{11}$ d. $y$ = $\frac{7}{25}$

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).