# Find the limit or show that it does not exist. \lim_{x\to\infty}=\frac{1-x^2}{x^3-x+1}

Find the limit or show that it does not exist.
$$\displaystyle\lim_{{{x}\to\infty}}={\frac{{{1}-{x}^{{2}}}}{{{x}^{{3}}-{x}+{1}}}}$$

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Charles Clute
Divide everything by the highes power of x, so $$\displaystyle{x}^{{3}}$$
$$\displaystyle\lim_{{{x}\to\infty}}{\frac{{{1}-{x}^{{2}}}}{{{x}^{{3}}-{x}+{1}}}}=\lim_{{{x}\to\infty}}{\frac{{{\frac{{{1}}}{{{x}^{{3}}}}}-{\frac{{{1}}}{{{x}}}}}}{{{1}-{\frac{{{1}}}{{{x}^{{2}}}}}+{\frac{{{1}}}{{{x}^{{3}}}}}}}}$$
Then as $$\displaystyle{x}\to\infty$$ terms $$\displaystyle{\frac{{{1}}}{{{x}^{{3}}}}},{\frac{{{1}}}{{{x}^{{2}}}}},{\frac{{{1}}}{{{x}}}}$$ approach 0.
$$\displaystyle{\frac{{{0}-{0}}}{{{1}-{0}+{0}}}}={\frac{{{0}}}{{{1}}}}={0}$$
$$\displaystyle\lim_{{{x}\to\infty}}{\frac{{{1}-{x}^{{2}}}}{{{x}^{{3}}-{x}+{1}}}}={0}$$