Exercise$\lim_{x\to\infty}\left(\frac{2x^3-2x^2+x-3}{x^3+2x^2-x+1}\right)$
Step-by-step Solution
1
As it's an indeterminate limit of type $\frac{\infty}{\infty}$, divide both numerator and denominator by the term of the denominator that tends more quickly to infinity (the term that, evaluated at a large value, approaches infinity faster). In this case, that term is
$\lim_{x\to\infty }\left(\frac{\frac{2x^3-2x^2+x-3}{x^3}}{\frac{x^3+2x^2-x+1}{x^3}}\right)$
2
Separate the terms of both fractions
$\lim_{x\to\infty }\left(\frac{\frac{2x^3}{x^3}+\frac{-2x^2}{x^3}+\frac{x}{x^3}+\frac{-3}{x^3}}{\frac{x^3}{x^3}+\frac{2x^2}{x^3}+\frac{-x}{x^3}+\frac{1}{x^3}}\right)$
Intermediate steps
$\lim_{x\to\infty }\left(\frac{2+\frac{-2x^2}{x^3}+\frac{x}{x^3}+\frac{-3}{x^3}}{1+\frac{2x^2}{x^3}+\frac{-x}{x^3}+\frac{1}{x^3}}\right)$
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4
Simplify the fraction by $x$
$\lim_{x\to\infty }\left(\frac{2+\frac{-2x^2}{x^3}+\frac{x}{x^3}+\frac{-3}{x^3}}{1+\frac{2x^2}{x^3}+\frac{-1}{x^{2}}+\frac{1}{x^3}}\right)$
5
Simplify the fraction by $x$
$\lim_{x\to\infty }\left(\frac{2+\frac{-2x^2}{x^3}+\frac{1}{x^{2}}+\frac{-3}{x^3}}{1+\frac{2x^2}{x^3}+\frac{-1}{x^{2}}+\frac{1}{x^3}}\right)$
Intermediate steps
6
Simplify the fraction by $x$
$\lim_{x\to\infty }\left(\frac{2+\frac{-2x^2}{x^3}+\frac{1}{x^{2}}+\frac{-3}{x^3}}{1+\frac{2}{x^{1}}+\frac{-1}{x^{2}}+\frac{1}{x^3}}\right)$
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Intermediate steps
7
Simplify the fraction by $x$
$\lim_{x\to\infty }\left(\frac{2+\frac{-2}{x^{1}}+\frac{1}{x^{2}}+\frac{-3}{x^3}}{1+\frac{2}{x^{1}}+\frac{-1}{x^{2}}+\frac{1}{x^3}}\right)$
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Intermediate steps
8
Any expression to the power of $1$ is equal to that same expression
$\lim_{x\to\infty }\left(\frac{2+\frac{-2}{x}+\frac{1}{x^{2}}+\frac{-3}{x^3}}{1+\frac{2}{x}+\frac{-1}{x^{2}}+\frac{1}{x^3}}\right)$
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Intermediate steps
9
Evaluate the limit $\lim_{x\to\infty }\left(\frac{2+\frac{-2}{x}+\frac{1}{x^{2}}+\frac{-3}{x^3}}{1+\frac{2}{x}+\frac{-1}{x^{2}}+\frac{1}{x^3}}\right)$ by replacing all occurrences of $x$ by $\infty $
$2$
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Final answer to the exercise $2$