How to find the asymptotes of #f(x)=( 21 x^2 ) / ( 3 x + 7)#?

Answer 1

The vertical asymptote is #x=-7/3#
The oblique asymptote is #y=7x-49/3#
There are no horizontal asymptotes

As we cannot divide by #0#, the vertical asymptote is #x=-7/3#

The degree of the numerator is #># than the degree of the numerator, so we expect a slant asymptote.

Let's do a long division

#color(white)(aaaa)##21x^2##color(white)(aaaaaaaaa)##∣##3x+7# #color(white)(aaaa)##21x^2+49x##color(white)(aaaa)##∣##7x-49/3# #color(white)(aaaaaaa)##0-49x# #color(white)(aaaaaaaaa)##-49x-343/3# #color(white)(aaaaaaaaaaa)##-0+343/3#

#:. f(x)=(7x-49/3)+(343/3)/(3x+7)# The oblique asymptote is #y=7x-49/3# graph{(y-21x^2/(3x+7))(y-7x+49/3)=0 [-169, 168.7, -84.7, 84.5]}

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.