how to find oblique asymptotes

HOW TO FIND SLANT ASYMPTOTE OF A FUNCTION

We will be able to find slant or oblique asymptote 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 :

Finding Slant or Oblique Asymptote of a Rational Function

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, or if thelargest exponent of the numerator is less than the largest exponent of the denominator, there is no slant asymptote.

Case 2 :

If thelargest exponent of the numerator is greater than the largest 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

Examples

Example 1 :

Find the slant or oblique asymptote of the graph of

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

Solution :

Step 1 :

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

Step 2 :

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

So, there is no slant asymptote.

Example 2 :

Find the slant or oblique asymptote of the graph of

f(x) = (x² + 2x - 3) / (x² - 5x + 6)

Solution :

Step 1 :

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

Step 2 :

Clearly, thelargest exponents of the numerator and the denominator are equal.

So, there is no slant asymptote.

Example 3 :

Find the slant or oblique asymptote of the graph of

f(x) = (x² + 3x + 2) / (x - 2)

Solution :

Step 1 :

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

Step 2 :

Clearly, thelargest exponent of the numerator is greater than the largest 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).

So, the equation of the slant asymptote is

y = x + 5

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