Explanation
The vertical asymptote
[tex]\begin{gathered} \mathrm{For\:rational\:functions,\:the\:vertical\:asymptotes\:are\:the\:undefined\:points,\:} \\ \mathrm{also\:known\:as\:the\:zeros\:of\:the\:denominator,\:of\:the\:simplified\:function.} \end{gathered}[/tex]
for the given function
[tex]T(x)=\frac{x^3}{x^4-81}[/tex]
According to the formula
The denominator will be undefined when
[tex]\begin{gathered} x^4-81=0 \\ x=\sqrt[4]{81} \\ x=3,\text{ x=-3} \\ x= \end{gathered}[/tex]
The vertical asymptotes are
[tex]x=-3,3[/tex]
For the horizontal function
[tex]\mathrm{If\:denominator's\:degree\:>\:numerator's\:degree,\:the\:horizontal\:asymptote\:is\:the\:x-axis:}\:y=0.[/tex]
Since the denominator degree is higher than the numerator
Then
The horizontal asymptote is
[tex]y=0[/tex]
For the oblique asymptote
Since the degree of the numerator is not one degree greater than the denominator, then there are no slant asymptotes.
There are no oblique asymptote
