We will be able to find horizontal asymptotes of a function, only if it is a rational function.

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

f(x) = g(x) / h(x)

**Rational Function - Example : **

Let f(x) be the given rational function. Compare the largest exponent of the numerator and denominator.

**Case 1 :**

If the largest exponents of the numerator and denominator are equal, equation of horizontal asymptote is

**y = a / b**

Here **a** and **b** are the coefficients of largest exponent terms at the numerator and denominator respectively.

**Case 2 :**

If the largest exponent of the numerator is less than the largest exponent of the denominator, equation of horizontal asymptote is

**y = o (or) x-axis**

**Case 3 :**

If the largest exponent of the numerator is greater than the largest exponent of the denominator, there is no horizontal asymptote and there is only slant asymptote or oblique.

**Example 1 :**

Find the equation of horizontal asymptote of the graph of

f(x) = 1 / (x + 6)

**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 :**

Find the equation of horizontal asymptote of the graph of

f(x) = (x^{2} + 2x - 3) / (x^{2} - 5x + 6)

**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 :**

Find the equation of horizontal asymptote of the graph of

f(x) = (x^{2} - 4) / (2x - 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.

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