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 :
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 the largest exponent of the numerator is less than the largest exponent of the denominator, there is no slant asymptote.
Case 2 :
If the largest 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
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) = (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 largest exponents of the numerator and the denominator are equal.
So, there is no slant asymptote.
Example 3 :
f(x) = (x^{2} + 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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