How do you find the Vertical, Horizontal, and Oblique Asymptote given #f(x)=( 21 x^2 ) / ( 3 x + 7)#?

1 Answer
Nov 8, 2016

Answer:

The vertical asymptote is #x=-7/3#
The oblique asymptote is #y=7x#
There is no horizontal asymptote

Explanation:

As you cannot divide by #0#, so #x=-7/3# is a vertical asymptote.

As the degree of the numerator #># the degre of the denominator,
therefore, we expect an oblique asymptote.

Let's do a long division
#21x^2##color(white)(aaaaaaaaa)##∣##3x+7#
#21x^2+49x##color(white)(aaaa)##∣##7x#
#color(white)(aa)##0-49x#

#:. (21x^2)/(3x+7)=7x-(49x)/(3x+7)#
So #y=7x# is an oblique asymptote
#lim_(x->+-oo)f(x)=lim_(x->+-oo)7x=+-oo#
So there is no horizontal asymptote
graph{(y-((21x^2)/(3x+7)))(y-7x)=0 [-277.3, 263.8, -233.8, 37]}