Is there a simpler way to calculate this limit?

  • Thread starter chrisb93
  • Start date
  • #1
8
0

Homework Statement



[itex]\lim_{x\to\infty} \frac{(1 - 2x^3)^4}{(x^4 - x^3+1)^3}[/itex]

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:

Answers and Replies

  • #2
NascentOxygen
Staff Emeritus
Science Advisor
9,244
1,072
Do you know the correct answer? Is it 16?
 
  • #4
NascentOxygen
Staff Emeritus
Science Advisor
9,244
1,072
Take the x3 factor out of the numerator, making it

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

Follow the same procedure for the denominator.
 
  • #5
8
0
Thanks so much, never thought of doing that to simplify limits.
 
  • #6
SammyS
Staff Emeritus
Science Advisor
Homework Helper
Gold Member
11,315
1,006
Alternatively: Multiply by [itex]\ \frac{\displaystyle\ \frac{1}{x^{12}}\ }{\displaystyle \frac{1}{x^{12}}}\ .[/itex]
 
  • #7
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
[tex]lim \frac{(-2)^4x^{12}}{x^{12}}=lim 16 = 16[/tex]

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

Related Threads on Is there a simpler way to calculate this limit?

Replies
25
Views
3K
Replies
5
Views
575
Replies
30
Views
2K
Replies
2
Views
2K
Replies
4
Views
1K
Replies
6
Views
1K
Replies
11
Views
1K
Replies
17
Views
2K
Replies
5
Views
1K
Replies
1
Views
1K
Top