How do I find the horizontal asymptotes in this (or any) limit? I attached the question as well as my work but I am lost with horizontal asymptotes so my work might be meaningless and I would appreciate any detailed explanation.
Thanks in advance!
How do I find the horizontal asymptotes in this (or any) limit? I attached the question as well as my work but I am lost with horizontal asymptotes so my work might be meaningless and I would appreciate any detailed explanation.
Thanks in advance!
The horizontal asymptotes of this function will be the vertical asymptotes of the inverse function.
So, the inverse function is defined by
$\displaystyle x = \frac{17y}{(y^4 + 1)^{\frac{1}{4}}}$
$\displaystyle x^4 = \frac{83521y^4}{y^4 + 1}$
$\displaystyle x^4 = 83521 - \frac{83521}{y^4 + 1}$
$\displaystyle x^4 - 83521 = - \frac{83521}{y^4 + 1}$
$\displaystyle \frac{1}{x^4 - 83521} = -\frac{y^4 + 1}{83521}$
$\displaystyle -\frac{83521}{x^4 - 83521} = y^4 + 1$
$\displaystyle -\frac{83521}{x^4 - 83521} - 1 = y^4$
$\displaystyle \left(-\frac{83521}{x^4 - 83521} - 1\right)^{\frac{1}{4}} = y$.
So the vertical asymptotes of the inverse function are $\displaystyle x = -17$ and $\displaystyle x = 17$.
Thus, the horizontal asymptotes of the original function are $\displaystyle y = -17$ and $\displaystyle y = 17$.