Question

Which function does not have a vertical asymptote? A) y=(x) /(1+x²) . B) y=(5x) /(1-x²) . C) y=(5x-1) /(x) . D) (5x) /(x+x²) .

Answer 1

V A are vertical lines that correspond to zeroes of the denominator of a rational function.
B ) Denominator: 1 - x² = 0
x² = 1,  x = 1, or x = - 1
C ) x = 0
D ) x + x² = 0
x ( 1 + x ) = 0,  x = 0 or x = - 1
And finally:
A ) 1 + x² ≠ 0,  x² ≠ - 1
Answer: function A ) does not have a vertical asymptote.

Answer 2

Function y=(x) /(1+x²) does not have a vertical asymptote.

What is a asymptote?

Asymptote is a line or curve that acts as the limit of another line or curve. An asymptote is a straight line that constantly approaches a given curve but does not meet at any infinite distance i.e. the distance between the curve and the straight line lends to zero when the points on the curve approach infinity.

What are the types of asymptotes?

There are two types of asymptotes a) vertical b) horizontal

When x moves to infinity or -infinity, the curve approaches some constant value b, and is called a Horizontal Asymptote.

When x approaches some constant value c from left or right, the curve moves towards infinity (i.e.,∞)  , or -infinity (i.e., -∞) and this is called Vertical Asymptote.

A) y=(x) /(1+x²)

Since we can see here the degree of the numerator is less than the denominator, therefore, the horizontal asymptote is located at y = 0.

B) y=(5x) /(1-x²)

Here y is not defined for x = 1.

y= 5(1)/(1-1²)

y= ∞

Therefore, the function y has a vertical asymptote at x = 1.

C) y=(5x-1) /(x)

Here y is not defined for x = 0

y= [5(0) -1] / 0

y=  ∞

Therefore, the function y has a vertical asymptote at x = 0

D) y=(5x) /(x+x²)

Here y is not defined for x = 0

y=[5(0)] / [0+0²]

y=  ∞

Therefore, the function y has a vertical asymptote at x = 0

Thus the function y=(x) /(1+x²) does not have a vertical asymptote.

To know more about vertical asymptote click here

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