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

elchatosarapage 2021-11-11 Answered
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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Expert Answer

Charles Clute
Answered 2021-11-12 Author has 7370 answers
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.
\(\displaystyle{\frac{{{0}-{0}}}{{{1}-{0}+{0}}}}={\frac{{{0}}}{{{1}}}}={0}\)
\(\displaystyle\lim_{{{x}\to\infty}}{\frac{{{1}-{x}^{{2}}}}{{{x}^{{3}}-{x}+{1}}}}={0}\)
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