What is the limit as x approaches infinity of (x^{100}+1/(x^3))/(pi-x^{97)} ?

The limit as x approaches infinity of (x^{100}+1/(x^3))/(pi-x^{97)} is -infinity

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limit as x approaches infinity of (x^{100}+1/(x^3))/(pi-x^{97)}

x \to \infty lim (\frac{x ^{100} + \frac{1}{x ^{3}}}{π - x ^{97}})

- \infty

Solution steps

x \to \infty lim (\frac{x ^{100} + \frac{1}{x ^{3}}}{π - x ^{97}})

Divide by highest denominator power:

\frac{x ^{3} + \frac{1}{x ^{100}}}{\frac{π}{x ^{97}} - 1}

= x \to \infty lim (\frac{x ^{3} + \frac{1}{x ^{100}}}{\frac{π}{x ^{97}} - 1})

x \to a lim [\frac{f ( x )}{g ( x )}] = \frac{lim _{x \to a} f ( x )}{lim _{x \to a} g ( x )}, x \to a lim g (x) \neq = 0

With the exception of indeterminate form

= \frac{lim _{x \to \infty} ( x ^{3} + \frac{1}{x ^{100}} )}{lim _{x \to \infty} ( \frac{π}{x ^{97}} - 1 )}

x \to \infty lim (x^{3} + \frac{1}{x ^{100}}) = \infty

x \to \infty lim (\frac{π}{x ^{97}} - 1) = - 1

= \frac{\infty}{- 1}

Apply Infinity Property:

\frac{\infty}{- c} = - \infty

= - \infty

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What is the limit as x approaches infinity of (x^{100}+1/(x^3))/(pi-x^{97)} ?
The limit as x approaches infinity of (x^{100}+1/(x^3))/(pi-x^{97)} is -infinity