Answer
$\lim_{x\to\infty}\dfrac{x^{4}}{1-x^{2}+x^{3}}$ does not exist
Work Step by Step
$\lim_{x\to\infty}\dfrac{x^{4}}{1-x^{2}+x^{3}}$
Try to evaluate the limit applying direct substitution:
$\lim_{x\to\infty}\dfrac{x^{4}}{1-x^{2}+x^{3}}=\dfrac{\infty^{4}}{1-\infty^{2}+\infty^{3}}=\dfrac{\infty}{\infty}$ Indeterminate form
Divide the numerator and the denominator of the function by $x^{4}$:
$\lim_{x\to\infty}\dfrac{x^{4}}{1-x^{2}+x^{3}}=\lim_{x\to\infty}\dfrac{\dfrac{x^{4}}{x^{4}}}{\dfrac{1-x^{2}+x^{3}}{x^{4}}}=...$
Simplify:
$...=\lim_{x\to\infty}\dfrac{1}{\dfrac{1}{x^{4}}-\dfrac{1}{x^{2}}+\dfrac{1}{x}}=...$
Apply direct substitution again:
$...=\dfrac{1}{\dfrac{1}{\infty^{4}}-\dfrac{1}{\infty^{2}}+\dfrac{1}{\infty}}=\dfrac{1}{0-0+0}=\dfrac{1}{0}=\infty$
