How do you find the vertical, horizontal and slant asymptotes of: #( x^2-2x)/(x^2-5x+4)#?

Question

Serenity Jones · Accepted Answer

vertical asymptotes at x = 1 and x = 4
horizontal asymptote at y = 1

The denominator of the function f(x) cannot be zero as this would make f(x) undefined. Equating the denominator to zero and solving gives the values that x cannot be and if the numerator is non-zero for these values then they are vertical asymptotes. solve : #x^2-5x+4=0rArr(x-1)(x-4)=0# #rArrx=1" and " x=4" are the asymptotes"# Horizontal asymptotes occur as #lim_(xto+-oo),f(x)toc" ( a constant)"# divide terms on numerator/denominator by the highest power of x, that is #x^2# #f(x)=(x^2/x^2-(2x)/x^2)/(x^2/x^2-(5x)/x^2+4/x^2)=(1-2/x)/(1-5/x+4/x^2)# as #xto+-oo,f(x)to(1-0)/(1-0+0)# #rArry=1" is the asymptote"# Slant asymptotes occur when the degree of the numerator > degree of the denominator. This is not the case here ( both of degree 2 ) Hence there are no slant asymptotes. graph{(x^2-2x)/(x^2-5x+4) [-10, 10, -5, 5]}

Lincoln Anderson · Answer

undefined To find the vertical asymptotes of the given rational function $ \frac{x^2 - 2x}{x^2 - 5x + 4} $:

1. Factor the numerator and denominator if possible.
2. Identify any values of $ x $ that make the denominator equal to zero. These values will give you vertical asymptotes.

To find the 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 $ 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 the slant asymptotes (if any):

1. If the degree of the numerator is exactly one more than the degree of the denominator, perform long division or polynomial division to find the quotient.
2. The quotient obtained will be the equation of the slant asymptote.

Apply these steps to the given function to find its vertical, horizontal, and slant asymptotes.