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Asymptote Formula

In geometry, asymptote is a line in such a way that the distance between the line and the curve tends to zero when they approaches to infinity. Mainly there are two types of asymptotes; horizontal and vertical asymptotes. The formula for horizontal and vertical asymptotes are,
Horizontal Asymptote:
  • Write down the given equation in y= form
  • Find out the ratio between the highest exponents of x in numerator and denominator

Note:
If the exponent of denominator is larger than that of numerator, horizontal asymptote, y=0. If the exponent of numerator is high, there is no horizontal asymptote.

Vertical Asymptote:

  • Set the denominator equal to zero and resolve the equation.

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Asymptote Problems

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Solved problems of asymptotes are given below:

Solved Examples

Question 1: Find out the horizontal and vertical asymptotes of  $\frac{2^{3}+5x+3}{3x^{3}-9}$ ?

Solution:
 
Given function is,
y = $\frac{2^{3}+5x+3}{3x^{3}-9}$

Highest exponent of numerator and denominator is 3. The coefficients are 2 and 3 respectively.

Horizontal asymptote is, $\frac{2}{3}$

Vertical asymptote can be calculated as,
3$x^{3}$ - 9 = 0
3$x^{3}$ = 9
$x^{3}$ = 3
x = $\sqrt[3]{3}$
x = 1.4422495
 

Question 2: Calculate the horizontal and vertical asymptotes of $\frac{x+2}{x^{2}-4}$ ?

Solution:
 
The given function is,
y = $\frac{x+2}{x^{2}-4}$

The exponent of denominator is higher than that of numerator. So horizontal asymptote is y = 0.
Vertical asymptote,
$x^{2}$ - 4 = 0
$x^{2}$ = 4
x = $\sqrt{4}$
x = 2
 

More topics in Asymptote Formula
Slant Asymptote Formula
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