**How to Find Slant Asymptote of a Function :**

In this section, we are going to learn, how to find slant asymptote (Oblique) of a function. .

And we will be able to find slant 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)

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

**Case 1 :**

If the highest exponent of the numerator and denominator are equal, or if the highest exponent of the
numerator is less than the highest exponent of the denominator, there is no slant asymptote.

**Case 2 :**

If the highest exponent of the numerator is greater than the highest exponent of the denominator by one, there is a slant asymptote.

To find slant asymptote, we have to use long division to divide the numerator by denominator.

When we divide so, let the quotient be (ax + b).

Then, the equation of the slant asymptote is

y = ax + b

**Example 1 :**

Find the equation of horizontal asymptote for the function given below.

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

**Solution : **

**Step 1 : **

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

**Step 2 :**

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

Hence, there is no slant asymptote.

**Example 2 :**

Find the equation of vertical asymptote for the function given below.

f(x) = (x² + 2x - 3) / (x² - 5x + 6)

**Solution : **

**Step 1 : **

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

**Step 2 :**

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

Hence, there is no slant asymptote.

**Example 3 :**

Find the equation of vertical asymptote for the function given below.

f(x) = (x² + 3x + 2) / (x - 2)

**Solution : **

**Step 1 : **

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

**Step 2 :**

Clearly, the exponent of the numerator is greater than the exponent of the denominator by one. So, there is a slant asymptote.

**Step 3 :**

To get the equation of the slant asymptote, we have to divide the numerator by the denominator using long division as given below.

**Step 3 :**

In the above long division, the quotient is (x + 5).

Hence, the equation of the slant asymptote is

y = x + 5

After having gone through the stuff given above, we hope that the students would have understood, "How to find slant asymptote of a function (Oblique)".

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