Limits to Infinity Calculator

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 Difficult Problems

Here, we show you a step-by-step solved example of limits to infinity. This solution was automatically generated by our smart calculator:

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

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

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

Separate the terms of both fractions

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

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

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

Simplify the fraction

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

Simplify the fraction by $x$

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

Simplify the fraction by $x$

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

Simplify the fraction by $x$

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

Subtract the values $3$ and $-2$

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

Simplify the fraction by $x$

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

Simplify the fraction by $x$

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

Subtract the values $3$ and $-2$

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

Simplify the fraction by $x$

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

Subtract the values $3$ and $-2$

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

Any expression to the power of $1$ is equal to that same expression

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

Simplify the fraction by $x$

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

Any expression to the power of $1$ is equal to that same expression

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

Any expression to the power of $1$ is equal to that same expression

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

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 $

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

Any expression divided by infinity is equal to zero

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

Infinity to the power of any positive number is equal to infinity, so $\infty ^3=\infty$

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

Infinity to the power of any positive number is equal to infinity, so $\infty ^3=\infty$

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

Infinity to the power of any positive number is equal to infinity, so $\infty ^3=\infty$

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

Infinity to the power of any positive number is equal to infinity, so $\infty ^3=\infty$

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

Infinity to the power of any positive number is equal to infinity, so $\infty ^3=\infty$

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

Infinity to the power of any positive number is equal to infinity, so $\infty ^{2}=\infty$

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

Infinity to the power of any positive number is equal to infinity, so $\infty ^3=\infty$

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

Combine fractions with common denominator $\infty $

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

Combine fractions with common denominator $\infty $

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

Combine fractions with common denominator $\infty $

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

Combine fractions with common denominator $\infty $

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

Combine fractions with common denominator $\infty $

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

Combine fractions with common denominator $\infty $

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

Combine fractions with common denominator $\infty $

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

Add the values $-2$ and $1$

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

Add the values $-1$ and $-3$

$2+\frac{-4}{\infty }$

Add the values $-2$ and $1$

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

Add the values $-1$ and $-3$

$2+\frac{-4}{\infty }$

Add the values $1$ and $-3$

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

Add the values $1$ and $-3$

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

Add the values $-1$ and $1$

$\frac{2+\frac{1}{\infty }+\frac{-3}{\infty }}{1+\frac{0}{\infty }}$

Combine fractions with common denominator $\infty $

$\frac{2+\frac{1-3}{\infty }}{1+\frac{0}{\infty }}$

Combine fractions with common denominator $\infty $

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

Combine fractions with common denominator $\infty $

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

Combine fractions with common denominator $\infty $

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

Combine fractions with common denominator $\infty $

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

Combine fractions with common denominator $\infty $

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

Combine fractions with common denominator $\infty $

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

Combine fractions with common denominator $\infty $

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

Add the values $-2$ and $1$

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

Add the values $-1$ and $-3$

$2+\frac{-4}{\infty }$

Add the values $-2$ and $1$

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

Add the values $-1$ and $-3$

$2+\frac{-4}{\infty }$

Add the values $1$ and $-3$

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

Add the values $1$ and $-3$

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

Add the values $-1$ and $1$

$\frac{2+\frac{1}{\infty }+\frac{-3}{\infty }}{1+\frac{0}{\infty }}$

Add the values $1$ and $-3$

$\frac{2+\frac{-2}{\infty }}{1+\frac{0}{\infty }}$

Any expression divided by infinity is equal to zero

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

Any expression divided by infinity is equal to zero

$\frac{2}{1}$

Divide $2$ by $1$

$2$

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 $

$2$

 Final answer to the exercise

$2$

Limits to Infinity Calculator

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