We will be able to find horizontal 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)
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, equation of horizontal asymptote is
y = a/b
Here a and b are the coefficients of largest exponent terms at the numerator and denominator respectively.
Case 2 :
If the largest exponent of the numerator is less than the largest exponent of the denominator, equation of horizontal asymptote is
y = o (or) x-axis
Case 3 :
If the largest exponent of the numerator is greater than the largest exponent of the denominator, there is no horizontal asymptote and there is only slant asymptote or oblique.
Find the equation of horizontal 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 largest exponent of the numerator is less than the largest exponent of the denominator.
So, equation of the horizontal asymptote is
y = 0 (or) x-axis
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 exponent of the numerator and the denominator are equal.
Step 3 :
Now, to get the equation of the horizontal asymptote, we have to divide the coefficients of largest exponent terms of the numerator and denominator.
So, equation of the horizontal asymptote is
y = 1/1
y = 1
Example 3 :
f(x) = (x^{2} - 4)/(2x - 3)
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.
Step 3 :
Because the largest exponent of the numerator is greater than the largest exponent of the denominator, there is no horizontal asymptote.
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