# Is there a simpler way to calculate this limit?

## Homework Statement

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

## Homework Equations

Not really applicable

## The Attempt at a Solution

After applying L'Hospital's rule 3 times I could just see it getting untidy and I couldn't see my and mistakes in my working so I checked against WolframAlpha which showed that L'Hospital's rule is needed 10 times to get to the solution which seems like a lot of work for a small question. Is there a more efficient method for solving this limit that I'm unaware of?

EDIT: Sorry if this should've been in the pre calculus section, I wasn't sure.

Last edited:

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NascentOxygen
Staff Emeritus
Do you know the correct answer? Is it 16?

Do you know the correct answer? Is it 16?
Yes it is

NascentOxygen
Staff Emeritus
Take the x3 factor out of the numerator, making it

(x3)4(1/x3 - 2)4

Follow the same procedure for the denominator.

Thanks so much, never thought of doing that to simplify limits.

SammyS
Staff Emeritus
Homework Helper
Gold Member
Alternatively: Multiply by $\ \frac{\displaystyle\ \frac{1}{x^{12}}\ }\frac{1}{x^{12}}}\$

if you're taking the limit as it goes to infinity you can drop any constants and only keep the highest power of x about since as you go to infinity those are going to be insignificant
doing this to your equation gives
$$lim \frac{(-2)^4x^{12}}{x^{12}}=lim 16 = 16$$

EDIT;
lol at wolframalpha applying L'hopitals rule 10 times