HOW TO FIND SLANT ASYMPTOTE OF A FUNCTION

Slant Asymptote (Oblique) :

A slant asymptote, just like a horizontal asymptote, guides the graph of a function for for large enough or small enough values of x, that is

x ----> ±∞

But it is a slanted line, that is, neither vertical nor horizontal.

We can 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 :

In each case, find the slant or oblique asymptote :

Example 1 :

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

So, there is no slant asymptote.

Example 2 :

f(x) = (x² + 2x - 3)/(x² - 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 largest exponents  of the numerator and the denominator are equal.

So, there is no slant asymptote.

Example 3 :

f(x) = (x² + 3x + 2)/(x - 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 1.

Step 2 :

Clearly, the largest 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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