How do you find vertical, horizontal and oblique asymptotes for #f(x) = (x)/( 4x^2+7x-2)#?

Question

Sadie Alexander · Accepted Answer

The Vertical Asymptotes are: #x=1/4 and x=-2#

The Horizontal Asymptote is #y=0 #

There is #No# Oblique Asymptote.

Finding the vertical asymptote of a rational function is by setting its denominator to zero that is #4x^2+7x-2=0# #rArr(4x-1)(x+2)=0# #4x-1=0# #rArr4x=1# #rArrx=1/4# #x+2=0# #rArrx=-2# The Vertical Asymptotes are: #color(red)(x=1/4 and x=-2)# Because the degree of numerator is less than the denominator So, #color(red)(y=0 )# is the Horizontal Asymptote There is #color(red)(No)# Oblique Asymptote.

Serenity Jones · Answer

undefined To find the vertical asymptotes, set the denominator equal to zero and solve for $x$. Any vertical asymptotes will occur at these values.

To find horizontal asymptotes:
1. Compare the degrees of the numerator and denominator.
2. If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is at $y = 0$.
3. If the degrees are equal, divide the leading coefficient of the numerator by the leading coefficient of the denominator to find the horizontal asymptote.

To find oblique asymptotes, perform polynomial long division or use synthetic division to divide the numerator by the denominator. The oblique asymptote is the quotient obtained.

Once you've found the vertical, horizontal, and oblique asymptotes, those equations represent the asymptotic behavior of the function as $x$ approaches positive or negative infinity.