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Limits to Infinity Calculator

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1

Solved example of limits to infinity

$\lim_{x\to\infty}\left(\frac{2x^3-2x^2+x-3}{x^3+2x^2-x+1}\right)$
2

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 $x^3$

$\lim_{x\to\infty }\left(\frac{\frac{2x^3-2x^2+x-3}{x^3}}{\frac{x^3+2x^2-x+1}{x^3}}\right)$

$\lim_{x\to\infty }\left(\frac{\frac{2x^3}{x^3}+\frac{-2x^2}{x^3}+\frac{x}{x^3}+\frac{-3}{x^3}}{\frac{x^3}{x^3}+\frac{2x^2}{x^3}+\frac{-x}{x^3}+\frac{1}{x^3}}\right)$

Simplify the fraction by $x$

$\lim_{x\to\infty }\left(\frac{\frac{2x^3}{x^3}+\frac{-2x^2}{x^3}+\frac{1}{x^{2}}+\frac{-3}{x^3}}{\frac{x^3}{x^3}+\frac{2x^2}{x^3}+\frac{-x}{x^3}+\frac{1}{x^3}}\right)$

Simplify the fraction by $x$

$\lim_{x\to\infty }\left(\frac{\frac{2x^3}{x^3}+\frac{-2x^2}{x^3}+\frac{1}{x^{2}}+\frac{-3}{x^3}}{\frac{x^3}{x^3}+\frac{2x^2}{x^3}+\frac{-1}{x^{2}}+\frac{1}{x^3}}\right)$

Simplify the fraction $\frac{2x^3}{x^3}$ by $x$

$\lim_{x\to\infty }\left(\frac{2+\frac{-2x^2}{x^3}+\frac{1}{x^{2}}+\frac{-3}{x^3}}{\frac{x^3}{x^3}+\frac{2x^2}{x^3}+\frac{-1}{x^{2}}+\frac{1}{x^3}}\right)$

Simplify the fraction $\frac{x^3}{x^3}$ by $x$

$\lim_{x\to\infty }\left(\frac{2+\frac{-2x^2}{x^3}+\frac{1}{x^{2}}+\frac{-3}{x^3}}{1+\frac{2x^2}{x^3}+\frac{-1}{x^{2}}+\frac{1}{x^3}}\right)$

Simplify the fraction by $x$

$\lim_{x\to\infty }\left(\frac{2+\frac{-2}{x}+\frac{1}{x^{2}}+\frac{-3}{x^3}}{1+\frac{2x^2}{x^3}+\frac{-1}{x^{2}}+\frac{1}{x^3}}\right)$

Simplify the fraction by $x$

$\lim_{x\to\infty }\left(\frac{2+\frac{-2}{x}+\frac{1}{x^{2}}+\frac{-3}{x^3}}{1+\frac{2}{x}+\frac{-1}{x^{2}}+\frac{1}{x^3}}\right)$
3

Separate the terms of both fractions

$\lim_{x\to\infty }\left(\frac{2+\frac{-2}{x}+\frac{1}{x^{2}}+\frac{-3}{x^3}}{1+\frac{2}{x}+\frac{-1}{x^{2}}+\frac{1}{x^3}}\right)$
4

The limit of the quotient of two functions is the quotient of their limits

$\frac{\lim_{x\to\infty }\left(2+\frac{-2}{x}+\frac{1}{x^{2}}+\frac{-3}{x^3}\right)}{\lim_{x\to\infty }\left(1+\frac{2}{x}+\frac{-1}{x^{2}}+\frac{1}{x^3}\right)}$
5

The limit of a polynomial function when $x$ tends to infinity is equal to the limit of it's highest degree term (the term that when i'ts evaluated at a high value, grows quickier to infinity), so it's solution is equivalent to calculating the limit of only the highest degree term

$\frac{\lim_{x\to\infty }\left(2\right)}{\lim_{x\to\infty }\left(1+\frac{2}{x}+\frac{-1}{x^{2}}+\frac{1}{x^3}\right)}$
6

The limit of a constant is just the constant

$\frac{2}{\lim_{x\to\infty }\left(1+\frac{2}{x}+\frac{-1}{x^{2}}+\frac{1}{x^3}\right)}$
7

The limit of a polynomial function when $x$ tends to infinity is equal to the limit of it's highest degree term (the term that when i'ts evaluated at a high value, grows quickier to infinity), so it's solution is equivalent to calculating the limit of only the highest degree term

$\frac{2}{\lim_{x\to\infty }\left(1\right)}$
8

The limit of a constant is just the constant

$2$

Final Answer

$2$

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