Slant Asymptotes

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Asymptote is a straight line which almost approaches the curve and doesn’t cross it at any finite points. There are three possible asymptotes for a given curve they are horizontal asymptote, vertical asymptote and slant asymptote. Slant asymptote is also called as oblique asymptote. A slant asymptote is where the numerator has greater degree than the degree of the denominator. The slant asymptote is the found by the method of long division where the numerator is the dividend and denominator is the divisor.   

Example 1: Find the slant asymptote of the curve
y = (x3 + 5x2)/x2?

Solution: Given is the curve y = (x3 + 5x2)/x2.

Here the degree of the numerator is 3 and the degree of the denominator is 2. The degree of the numerator is greater than the degree of the denominator.

The numerator can be factored as (x3 + 5x2) = x2 (x +5).

The denominator is x2 dividing gives x2 (x +5)/ x2 = x +5.

Hence the slant asymptote is y = x+5.
 
Example 2: Find the slant asymptote of the curve
y = (x2 + 3x + 2)/(x+2)?

Solution: Given is the curve y = (x2 + 3x + 2)/(x+2)

Here the degree of the numerator is 2 and the degree of the denominator is 1. The degree of the numerator is greater than the degree of the denominator.

The numerator can be factored as (x2 + 3x + 2) = (x + 2) (x +1).

The denominator is (x +2) dividing gives (x +2) (x +1)/ (x + 2) = x +1.

Hence the slant asymptote is y = x +1.
                                       

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