Does cancellation impact vertical asymptotes? Question: Let r(x)=(x2+x)/((x+1)(2x−4)). Does the graph has x=1 as one of its asymptotes? My reasoning: (x2+x)/((x+1)(2x−4))=x(x+1)/((x+1)(2x−4))=x/(2x−4) and so, it cannot have x=−1 as one of its asymptotes. However, what if I don't calcel and then say that −1 is a vertical asymptote? Will I be wrong?

glyperezrl 2022-07-17 Answered
Does cancellation impact vertical asymptotes?
Question: Let r ( x ) = ( x 2 + x ) ( x + 1 ) ( 2 x 4 ) . Does the graph has x=1 as one of its asymptotes?
Answer: No.
My reasoning: ( x 2 + x ) ( x + 1 ) ( 2 x 4 ) = x ( x + 1 ) ( x + 1 ) ( 2 x 4 ) = x ( 2 x 4 ) and so, it cannot have x=−1 as one of its asymptotes.
However, what if I don't calcel and then say that −1 is a vertical asymptote? Will I be wrong?
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Answers (1)

Julianna Bell
Answered 2022-07-18 Author has 19 answers
You are correct saying that the function
r ( x ) = ( x 2 + x ) ( x + 1 ) ( 2 x 4 )
has not an asymptote for x=−1 since:
lim x 1 ( x 2 + x ) ( x + 1 ) ( 2 x 4 ) = lim x 1 x ( x + 1 ) ( x + 1 ) ( 2 x 4 ) = 1 6
but this function is not defined for x=−1 so its graph has an ''hole'' at the point ( 1 , 1 6 ) and the functios is not the same as
y = x ( 2 x 4 )

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