HOW TO FIND HORIZONTAL ASYMPTOTE OF A FUNCTION

We can find horizontal asymptotes of a function, only if it is a rational function.

That is, the function has to be in the form of

Rational Function - Example :

In each case, find the equation of horizontal asymptote.

Example 1 :

Solution :

Step 1 :

In the given rational function, the largest exponent of the numerator is 0 and the largest exponent of the denominator is 1.

Step 2 :

Clearly largest exponent of the numerator is less than the largest exponent of the denominator.

So, equation of the horizontal asymptote is

y = 0 (or) x-axis

Example 2 :

Solution :

Step 1 :

In the given rational function, the largest exponent of the numerator is 2 and the largest exponent of the denominator is 2.

Step 2 :

Clearly, the exponent of the numerator and the denominator are equal.

Step 3 :

Now, to get the equation of the horizontal asymptote, we have to divide the coefficients of largest exponent terms of the numerator and denominator.

So, equation of the horizontal asymptote is

y = ¹⁄₁

y = 1

Example 3 :

Solution :

Step 1 :

In the given rational function, the largest exponent of the numerator is 2 and the largest exponent of the denominator is 1.

Step 2 :

Clearly, the largest exponent of the numerator is greater than the largest exponent of the denominator.

Step 3 :

Because the largest exponent of the numerator is greater than the largest exponent of the denominator, there is no horizontal asymptote.

Example 4 :

Solution :

Step 1 :

Convert the mixed expression on the right side to rational expression.

Step 2 :

In the given rational function, the largest exponent of the numerator is 1 and the largest exponent of the denominator is also 1.

Step 3 :

Clearly, the exponent of the numerator and the denominator are equal.

Step 4 :

Now, to get the equation of the horizontal asymptote, we have to divide the coefficients of largest exponent terms of the numerator and denominator.

So, equation of the horizontal asymptote is

y = ⁶⁄₁

y = 6

Example 5 :

y = e^x

Solution :

When x ---> -∞, 

When x ---> +∞, 

When x ---> -∞, e^x ---> 0.

When x ---> +∞, e^x ---> +∞.

When x tends to -∞, e^x tends to a finite value 0. 

That is,

x ---> -∞, y ---> 0

Therefore, the horizontal asymptote is

y = 0

Example 6 :

y = e^-x

Solution :

When x ---> -∞, 

When x ---> +∞, 

When x ---> -∞, e^-x ---> +∞.

When x ---> +∞, e^-x ---> 0.

When x tends to +∞, e^-x tends to a finite value 0. 

That is,

x ---> +∞, y ---> 0

Therefore, the horizontal asymptote is

y = 0

Example 7 :

Solution :

When x ---> -∞, 

When x ---> +∞, 

When x ---> ±∞, 

When x tends to ±∞, e^¹⁄ₓ tends to a finite value 1. 

That is,

x ---> ±∞, y ---> 1

Therefore, the horizontal asymptote is

y = 1

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