How do you find the vertical, horizontal or slant asymptotes for #f(x)= (x+5)/(x+3)#?

1 Answer
Jul 19, 2016

Answer:

vertical asymptote x = -3
horizontal asymptote y = 1

Explanation:

The denominator of f(x) cannot be zero as this is undefined. Equating the denominator to zero and solving gives the value that x cannot be and if the numerator is non-zero for this value then it is a vertical asymptote.

solve: x + 3 = 0 → x = -3 is the asymptote

Horizontal asymptotes occur as

#lim_(xto+-oo),f(x)toc" (a constant)"#

divide terms on numerator/denominator by x

#(x/x+5/x)/(x/x+3/x)=(1+5/x)/(1+3/x)#

as #xto+-00,f(x)to(1+0)/(1+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 1) Hence there are no slant asymptotes.
graph{(x+5)/(x+3) [-10, 10, -5, 5]}