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Find the limit of $\frac{2x^3-2x^2+x-3}{x^3+2x^2-x+1}$ as $x$ approaches $\infty $

Step-by-step Solution

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Solution

Final Answer

$2$
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Step-by-step Solution

Specify the solving method

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

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

Simplify the fraction

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

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

Simplify the fraction by $x$

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

Simplify the fraction by $x$

$\lim_{x\to\infty }\left(\frac{2+\frac{-2}{x}+\frac{1}{x^{2}}+\frac{-3}{x^3}}{1+\frac{2}{x}+\frac{-1}{x^{2}}+\frac{1}{x^3}}\right)$
8

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 $

$\frac{2+\frac{-2}{\infty }+\frac{1}{\infty ^{2}}+\frac{-3}{\infty ^3}}{1+\frac{2}{\infty }+\frac{-1}{\infty ^{2}}+\frac{1}{\infty ^3}}$
9

Any expression divided by infinity is equal to zero

$\frac{2+\frac{-2}{\infty }+\frac{1}{\infty ^{2}}+\frac{-3}{\infty ^3}}{1+0+\frac{-1}{\infty ^{2}}+\frac{1}{\infty ^3}}$
10

Add the values $1$ and $0$

$\frac{2+\frac{-2}{\infty }+\frac{1}{\infty ^{2}}+\frac{-3}{\infty ^3}}{1+\frac{-1}{\infty ^{2}}+\frac{1}{\infty ^3}}$
11

Any expression divided by infinity is equal to zero

$\frac{2+0+\frac{1}{\infty ^{2}}+\frac{-3}{\infty ^3}}{1+\frac{-1}{\infty ^{2}}+\frac{1}{\infty ^3}}$
12

Add the values $2$ and $0$

$\frac{2+\frac{1}{\infty ^{2}}+\frac{-3}{\infty ^3}}{1+\frac{-1}{\infty ^{2}}+\frac{1}{\infty ^3}}$
13

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

$2\cdot \infty -2\cdot \infty +\infty -3$
14

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

$\infty +2\cdot \infty - \infty +1$
15

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

$2+\frac{-2}{\infty }+\frac{1}{\infty }+\frac{-3}{\infty }$
16

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

$2+\frac{-2}{\infty }+\frac{1}{\infty }+\frac{-3}{\infty }$
17

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

$2+\frac{1}{\infty }+\frac{-3}{\infty }$
18

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

$\frac{2+\frac{1}{\infty ^{2}}+\frac{-3}{\infty ^3}}{1+\frac{-1}{\infty }+\frac{1}{\infty }}$
19

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

$\frac{2+\frac{1}{\infty }+\frac{-3}{\infty ^3}}{1+\frac{-1}{\infty }+\frac{1}{\infty }}$
20

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

$\frac{2+\frac{1}{\infty }+\frac{-3}{\infty }}{1+\frac{-1}{\infty }+\frac{1}{\infty }}$
21

Any expression divided by infinity is equal to zero

$\frac{2+\frac{1}{\infty }+\frac{-3}{\infty }}{1+0+\frac{1}{\infty }}$
22

Add the values $1$ and $0$

$\frac{2+\frac{1}{\infty }+\frac{-3}{\infty }}{1+\frac{1}{\infty }}$
23

Any expression divided by infinity is equal to zero

$\frac{2+\frac{1}{\infty }+\frac{-3}{\infty }}{1+0}$
24

Add the values $1$ and $0$

$\frac{2+\frac{1}{\infty }+\frac{-3}{\infty }}{1}$
25

Any expression divided by one ($1$) is equal to that same expression

$2+\frac{1}{\infty }+\frac{-3}{\infty }$
26

Any expression divided by infinity is equal to zero

$2+0+\frac{-3}{\infty }$
27

Add the values $2$ and $0$

$2+\frac{-3}{\infty }$
28

Any expression divided by infinity is equal to zero

$2+0$
29

Add the values $2$ and $0$

$2$

Final Answer

$2$

Exact Numeric Answer

$2$

Explore different ways to solve this problem

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Plotting: $\frac{2x^3-2x^2+x-3}{x^3+2x^2-x+1}$

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a
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d
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g
m
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u
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w
x
y
z
.
(◻)
+
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×
◻/◻
/
÷
◻2
◻◻
√◻
√

◻√◻
◻√
∞
e
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log
log◻
lim
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|◻|
θ
=
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sin
cos
tan
cot
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csc

asin
acos
atan
acot
asec
acsc

sinh
cosh
tanh
coth
sech
csch

asinh
acosh
atanh
acoth
asech
acsch

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