Find all asymptotes of a function: f(x)=log(x^2−4) Domain: xin(−infty,−2)cup(2,infty) Vertical asymptotes are x=−2 (left) and x=2(right): limx→ −2−log(x2−4)=−infty

Mbalisikerc 2022-07-18 Answered
Find all asymptotes of a function
Find all asymptotes of a function:
f ( x ) = log ( x 2 4 )
Domain: x ( , 2 ) ( 2 , )
Vertical asymptotes are x=−2 (left) and x=2(right):
lim x   2 log ( x 2 4 ) =
lim x   2 + log ( x 2 4 ) =
I calculate the limits in +/- infinity:
lim x   + log ( x 2 4 ) = +
lim x   log ( x 2 4 ) = +
So I'm looking for the oblique asymptotes of a form y=Ax+B:
A + = lim x   + log ( x 2 4 ) x = lim x   + 2 x x 2 4 1 = 0
B + = lim x   + log ( x 2 4 ) A + x = +
The same for . How should I interpret this? There are no oblique asymptotes?
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Answers (1)

umshikepl
Answered 2022-07-19 Author has 11 answers
Yes, there are no oblique asymptotes. In general there are when A and B are finite real numbers. In the particular case A=0 and B is a constant, then you find a horizontal asymptote.
The computations are correct.

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