馃憠 Try now NerdPal! Our new math app on iOS and Android
Calculators Topics Solving Methods Step Checker
Book solutions
Algebra Baldor
Go Premium
ENG • ESP
Topics

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

Go!
Math mode
Text mode
Go!
1
2
3
4
5
6
7
8
9
0
a
b
c
d
f
g
m
n
u
v
w
x
y
z
.
(◻)
+
-
×
◻/◻
/
÷
◻2
◻◻
√◻
√

◻√◻
◻√
∞
e
π
ln
log
log◻
lim
d/dx
D□x
∫
∫◻
|◻|
θ
=
>
<
>=
<=
sin
cos
tan
cot
sec
csc

asin
acos
atan
acot
asec
acsc

sinh
cosh
tanh
coth
sech
csch

asinh
acosh
atanh
acoth
asech
acsch

Solution

Final Answer

$2$
Got another answer? Verify it here!

Step-by-step Solution

Problem to solve:

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

Specify the solving method

As a variable goes to infinity, the expression $2x^3-2x^2+x-3$ will behave the same way that it's largest power behaves

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

As a variable goes to infinity, the expression $x^3+2x^2-x+1$ will behave the same way that it's largest power behaves

$\frac{2x^3}{x^3}$

Plug in the value $\infty $ into the limit

$\frac{2\infty ^3}{\infty ^3}$

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

$\frac{2\cdot \infty }{\infty ^3}$

Any expression multiplied by infinity tends to infinity

$\frac{\infty }{\infty ^3}$

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

$\frac{\infty }{\infty }$
1

If we directly evaluate the limit $\lim_{x\to \infty }\left(\frac{2x^3-2x^2+x-3}{x^3+2x^2-x+1}\right)$ as $x$ tends to $\infty $, we can see that it gives us an indeterminate form

$\frac{\infty }{\infty }$
2

We can solve this limit by applying L'H么pital's rule, which consists of calculating the derivative of both the numerator and the denominator separately

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

Find the derivative of the numerator

$\frac{d}{dx}\left(2x^3-2x^2+x-3\right)$

The derivative of a sum of two or more functions is the sum of the derivatives of each function

$\frac{d}{dx}\left(2x^3\right)+\frac{d}{dx}\left(-2x^2\right)+\frac{d}{dx}\left(x\right)+\frac{d}{dx}\left(-3\right)$

The derivative of the constant function ($-3$) is equal to zero

$\frac{d}{dx}\left(2x^3\right)+\frac{d}{dx}\left(-2x^2\right)+\frac{d}{dx}\left(x\right)$

The derivative of the linear function is equal to $1$

$\frac{d}{dx}\left(2x^3\right)+\frac{d}{dx}\left(-2x^2\right)+1$

The derivative of a function multiplied by a constant ($2$) is equal to the constant times the derivative of the function

$2\frac{d}{dx}\left(x^3\right)+\frac{d}{dx}\left(-2x^2\right)+1$

The power rule for differentiation states that if $n$ is a real number and $f(x) = x^n$, then $f'(x) = nx^{n-1}$

$6x^{2}+\frac{d}{dx}\left(-2x^2\right)+1$

The derivative of a function multiplied by a constant ($-2$) is equal to the constant times the derivative of the function

$6x^{2}-2\frac{d}{dx}\left(x^2\right)+1$

The power rule for differentiation states that if $n$ is a real number and $f(x) = x^n$, then $f'(x) = nx^{n-1}$

$6x^{2}-4x+1$

Find the derivative of the denominator

$\frac{d}{dx}\left(x^3+2x^2-x+1\right)$

The derivative of a sum of two or more functions is the sum of the derivatives of each function

$\frac{d}{dx}\left(x^3\right)+\frac{d}{dx}\left(2x^2\right)+\frac{d}{dx}\left(-x\right)+\frac{d}{dx}\left(1\right)$

The derivative of the constant function ($1$) is equal to zero

$\frac{d}{dx}\left(x^3\right)+\frac{d}{dx}\left(2x^2\right)+\frac{d}{dx}\left(-x\right)$

The derivative of the linear function times a constant, is equal to the constant

$\frac{d}{dx}\left(x^3\right)+\frac{d}{dx}\left(2x^2\right)-\frac{d}{dx}\left(x\right)$

The derivative of a function multiplied by a constant ($2$) is equal to the constant times the derivative of the function

$\frac{d}{dx}\left(x^3\right)+2\frac{d}{dx}\left(x^2\right)-\frac{d}{dx}\left(x\right)$

The power rule for differentiation states that if $n$ is a real number and $f(x) = x^n$, then $f'(x) = nx^{n-1}$

$3x^{2}+2\frac{d}{dx}\left(x^2\right)-\frac{d}{dx}\left(x\right)$

The derivative of the linear function is equal to $1$

$3x^{2}+2\frac{d}{dx}\left(x^2\right)-1$

The power rule for differentiation states that if $n$ is a real number and $f(x) = x^n$, then $f'(x) = nx^{n-1}$

$3x^{2}+4x-1$
3

After deriving both the numerator and denominator, the limit results in

$\lim_{x\to\infty }\left(\frac{6x^{2}-4x+1}{3x^{2}+4x-1}\right)$

As a variable goes to infinity, the expression $6x^{2}-4x+1$ will behave the same way that it's largest power behaves

$\frac{6x^{2}}{3x^{2}+4x-1}$

As a variable goes to infinity, the expression $3x^{2}+4x-1$ will behave the same way that it's largest power behaves

$\frac{6x^{2}}{3x^{2}}$

Plug in the value $\infty $ into the limit

$\frac{6\infty ^{2}}{3\infty ^{2}}$

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

$\frac{6\cdot \infty }{3\infty ^{2}}$

Any expression multiplied by infinity tends to infinity

$\frac{\infty }{3\infty ^{2}}$

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

$\frac{\infty }{3\cdot \infty }$

Any expression multiplied by infinity tends to infinity

$\frac{\infty }{\infty }$
4

If we directly evaluate the limit $\lim_{x\to \infty }\left(\frac{6x^{2}-4x+1}{3x^{2}+4x-1}\right)$ as $x$ tends to $\infty $, we can see that it gives us an indeterminate form

$\frac{\infty }{\infty }$
5

We can solve this limit by applying L'H么pital's rule, which consists of calculating the derivative of both the numerator and the denominator separately

$\lim_{x\to \infty }\left(\frac{\frac{d}{dx}\left(6x^{2}-4x+1\right)}{\frac{d}{dx}\left(3x^{2}+4x-1\right)}\right)$

Find the derivative of the numerator

$\frac{d}{dx}\left(6x^{2}-4x+1\right)$

The derivative of a sum of two or more functions is the sum of the derivatives of each function

$\frac{d}{dx}\left(6x^{2}\right)+\frac{d}{dx}\left(-4x\right)+\frac{d}{dx}\left(1\right)$

The derivative of the constant function ($1$) is equal to zero

$\frac{d}{dx}\left(6x^{2}\right)+\frac{d}{dx}\left(-4x\right)$

The derivative of the linear function times a constant, is equal to the constant

$\frac{d}{dx}\left(6x^{2}\right)-4\frac{d}{dx}\left(x\right)$

The derivative of a function multiplied by a constant ($6$) is equal to the constant times the derivative of the function

$6\frac{d}{dx}\left(x^{2}\right)-4\frac{d}{dx}\left(x\right)$

The power rule for differentiation states that if $n$ is a real number and $f(x) = x^n$, then $f'(x) = nx^{n-1}$

$12x-4\frac{d}{dx}\left(x\right)$

The derivative of the linear function is equal to $1$

$12x-4$

Find the derivative of the denominator

$\frac{d}{dx}\left(3x^{2}+4x-1\right)$

The derivative of a sum of two or more functions is the sum of the derivatives of each function

$\frac{d}{dx}\left(3x^{2}\right)+\frac{d}{dx}\left(4x\right)+\frac{d}{dx}\left(-1\right)$

The derivative of the constant function ($-1$) is equal to zero

$\frac{d}{dx}\left(3x^{2}\right)+\frac{d}{dx}\left(4x\right)$

The derivative of the linear function times a constant, is equal to the constant

$\frac{d}{dx}\left(3x^{2}\right)+4\frac{d}{dx}\left(x\right)$

The derivative of a function multiplied by a constant ($3$) is equal to the constant times the derivative of the function

$3\frac{d}{dx}\left(x^{2}\right)+4\frac{d}{dx}\left(x\right)$

The power rule for differentiation states that if $n$ is a real number and $f(x) = x^n$, then $f'(x) = nx^{n-1}$

$6x+4\frac{d}{dx}\left(x\right)$

The derivative of the linear function is equal to $1$

$6x+4$

Factor the numerator by $2$

$\lim_{x\to\infty }\left(\frac{2\left(6x-2\right)}{6x+4}\right)$

Factor the denominator by $2$

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

Cancel the fraction's common factor $2$

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

After deriving both the numerator and denominator, the limit results in

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

Plug in the value $\infty $ into the limit

$\frac{6\cdot \infty -2}{3\cdot \infty +2}$

Any expression multiplied by infinity tends to infinity

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

Infinity plus any algebraic expression is equal to infinity

$\frac{\infty }{3\cdot \infty +2}$

Any expression multiplied by infinity tends to infinity

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

Infinity plus any algebraic expression is equal to infinity

$\frac{\infty }{\infty }$
7

If we directly evaluate the limit $\lim_{x\to \infty }\left(\frac{6x-2}{3x+2}\right)$ as $x$ tends to $\infty $, we can see that it gives us an indeterminate form

$\frac{\infty }{\infty }$
8

We can solve this limit by applying L'H么pital's rule, which consists of calculating the derivative of both the numerator and the denominator separately

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

Find the derivative of the numerator

$\frac{d}{dx}\left(6x-2\right)$

The derivative of a sum of two or more functions is the sum of the derivatives of each function

$\frac{d}{dx}\left(6x\right)+\frac{d}{dx}\left(-2\right)$

The derivative of the constant function ($-2$) is equal to zero

$\frac{d}{dx}\left(6x\right)$

The derivative of the linear function times a constant, is equal to the constant

$6\frac{d}{dx}\left(x\right)$

The derivative of the linear function is equal to $1$

$6$

Find the derivative of the denominator

$\frac{d}{dx}\left(3x+2\right)$

The derivative of a sum of two or more functions is the sum of the derivatives of each function

$\frac{d}{dx}\left(3x\right)+\frac{d}{dx}\left(2\right)$

The derivative of the constant function ($2$) is equal to zero

$\frac{d}{dx}\left(3x\right)$

The derivative of the linear function times a constant, is equal to the constant

$3\frac{d}{dx}\left(x\right)$

The derivative of the linear function is equal to $1$

$3$

Divide $6$ by $3$

$\lim_{x\to\infty }\left(2\right)$
9

After deriving both the numerator and denominator, the limit results in

$\lim_{x\to\infty }\left(2\right)$
10

The limit of a constant is just the constant

$2$

Final Answer

$2$

Exact Numeric Answer

$2$

Explore different ways to solve this problem

Solving a math problem using different methods is important because it enhances understanding, encourages critical thinking, allows for multiple solutions, and develops problem-solving strategies. Read more

Limits by Direct SubstitutionLimits by L'H么pital's ruleLimits by factoringLimits by rationalizing

Give us your feedback!

SnapXam A2
Answer Assistant

beta
Got a different answer? Verify it!

Go!
1
2
3
4
5
6
7
8
9
0
a
b
c
d
f
g
m
n
u
v
w
x
y
z
.
(◻)
+
-
×
◻/◻
/
÷
◻2
◻◻
√◻
√

◻√◻
◻√
∞
e
π
ln
log
log◻
lim
d/dx
D□x
∫
∫◻
|◻|
θ
=
>
<
>=
<=
sin
cos
tan
cot
sec
csc

asin
acos
atan
acot
asec
acsc

sinh
cosh
tanh
coth
sech
csch

asinh
acosh
atanh
acoth
asech
acsch

How to improve your answer:

Supercharge your math learning

Use your academic email to get more benefits.
By signing up, you agree to SnapXam's Terms and Privacy Policy.

Already have an account? Login here
Forgot password?

Supercharge your math learning

Use your academic email to get more benefits.


Register here if you don't have an account.
Forgot password?

Invest in your Education!

Help us make you learn faster

Complete step-by-step math solutions. No ads.

Including multiple solving methods.

Support for more than 100 math topics.

Premium access on our iOS and Android app.

Join 500k+ students in problem solving.

Subscription. Cancel anytime.
Have a promo code?
Pay $39.97 USD securely with your payment method.
Please hold while your payment is being processed.
Create an Account
3-Month Special Plan
One-time payment of $2.97 USD.
Without automatic renewal.
Create an Account