How do you find the asymptotes for #y= (7x-2) /( x^2-3x-4)#?

Question

How do you find the asymptotes for #y=  (7x-2) /( x^2-3x-4)#?

Anna Allen · Accepted Answer

The asymptotes of any expression are found by defining what happens to the expression when #x -> oo# or #x-> - oo# or when #y->oo# The asymptotes of any expression are found by defining what happens to the expression when #x -> oo# or #x-> - oo# or when #y->oo# In this case # y =(7x+2)/(x^2 - 3x - 4)# or # (7x-2)/((x-4)(x+1))# Hence when #x ->4# or #x -> -1# then #y -> oo#. Therefore there are vertical asymptotes at #x =4# and at #x=-1# #lim _( x-> +-oo) (7x-2)/(x^2-3x-4) = lim_(x->+-oo) 7/x =0#

Liam Brennan · Answer

undefined To find the asymptotes of the rational function $y = \frac{{7x - 2}}{{x^2 - 3x - 4}}$, follow these steps:

1. Determine the degree of the numerator and the denominator.
2. If the degree of the numerator is less than the degree of the denominator, there is a horizontal asymptote at $y = 0$.
3. If the degree of the numerator equals the degree of the denominator, divide the leading coefficients to find the horizontal asymptote.
4. If the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote. Instead, perform polynomial long division to find any possible slant (oblique) asymptote.
5. To find the vertical asymptotes, set the denominator equal to zero and solve for $x$. The vertical asymptotes occur where the function is undefined.
6. If there are any common factors between the numerator and the denominator, simplify the function before identifying asymptotes.

Follow these steps to find the asymptotes for the given function.