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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 $x^3$

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

Separate the terms of both fractions

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

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

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

Infinity to the power of any positive number is equal to infinity

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

Any expression divided by infinity is equal to zero

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

Infinity to the power of any positive number is equal to infinity

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

Any expression divided by infinity is equal to zero

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

Any expression divided by infinity is equal to zero

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

Simplifying

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

Evaluate the limit by replacing all occurrences of $x$ by $\infty $

$\frac{2}{1+\frac{2}{\infty }+\frac{-1}{\infty ^{2}}+\frac{1}{\infty ^3}}$

Infinity to the power of any positive number is equal to infinity

$\frac{2}{1+\frac{2}{\infty }+\frac{-1}{\infty }+\frac{1}{\infty ^3}}$

Any expression divided by infinity is equal to zero

$\frac{2}{1+\frac{-1}{\infty }+\frac{1}{\infty ^3}}$

Infinity to the power of any positive number is equal to infinity

$\frac{2}{1+\frac{-1}{\infty }+\frac{1}{\infty }}$

Any expression divided by infinity is equal to zero

$\frac{2}{1+\frac{1}{\infty }}$

Any expression divided by infinity is equal to zero

$2$
7

Simplifying

$2$

Final Answer

$2$

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