Find all asymptotes of a functionFind 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?

Question

Find all asymptotes of a functionFind 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&#039;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?

umshikepl · Accepted Answer

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.

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

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