Is there a simpler way to calculate this limit?

[chrisb93]

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.

 

[NascentOxygen]

Do you know the correct answer? Is it 16?

 

[chrisb93]

Do you know the correct answer? Is it 16?

Yes it is

 

[NascentOxygen]

Take the x³ factor out of the numerator, making it

(x³)⁴(1/x³ - 2)⁴

Follow the same procedure for the denominator.

 

[chrisb93]

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

 

[SammyS]

Alternatively: Multiply by $\ \frac{\displaystyle\ \frac{1}{x^{12}}\ }{\displaystyle \frac{1}{x^{12}}}\ .$

 

[genericusrnme]

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