How do you find vertical, horizontal and oblique asymptotes for #(6x + 6) / (3x^2 + 1)#?
1 Answer
Explanation:
#"let "f(x)=(6x+6)/(3x^2+1)# The denominator of f(x) cannot be zero as this would make f(x) 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 "3x^2+1=0rArrx^2=-1/3#
#"This has no real solutions hence there are no vertical"#
#"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)=((6x)/x^2+6/x^2)/((3x^2)/x^2+1/x^2)=(6/x+6/x^2)/(3+1/x^2)#
#"as "xto+-oo,f(x)to(0+0)/(3+0)#
#y=0" is the asymptote"#
#"Oblique asymptotes occur when the degree of the"#
#"numerator is greater than the degree of the denominator. "#
#"This is not the case here hence no oblique asymptotes"#
graph{(6x+6)/(3x^2+1) [-10, 10, -5, 5]}
