Finding the limit as x tends to infinity

**Glitch** · September 9th 2010, 12:52 AM

The question:
Determine the limiting behaviour of $\frac{ln(x^3+1)}{ln(x^2+1)}$ as x -> infinity.

My attempt:
I noticed that both the numerator and denominator approach infinity, making the limit indeterminate. Thus, I tried using L'Hopitals rule. However, in doing so, I can't seem to get an answer even after several differentiations. It seems to constantly become 0/0. Am I attempting this the right way?

Thanks!

**Prove It** · September 9th 2010, 01:00 AM

Using L'Hospital...

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

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

$= \lim_{x \to \infty}\frac{3x^4 + 3x^2}{2x^4 + 2x}$ .

This is still $\frac{\infty}{\infty}$ so use L'Hospital again...

$\lim_{x \to \infty}\frac{3x^4 + 3x^2}{2x^4 + 2x} = \lim_{x \to \infty}\frac{12x^3 + 6x}{8x^3 + 2}$ .

This is still $\frac{\infty}{\infty}$ so use L'Hospital again...

$\lim_{x \to \infty}\frac{12x^3 + 6x}{8x^3 + 2} = \lim_{x\to\infty}\frac{36x^2 + 6}{24x^2}$ .

This is still $\frac{\infty}{\infty}$ so use L'Hospital again

$\lim_{x \to \infty}\frac{36x^2 + 6}{24x^2} = \lim_{x \to \infty}\frac{72x}{48x}$

$= \lim_{x \to \infty}\frac{3}{2}$

$= \frac{3}{2}$ .

**Glitch** · September 9th 2010, 01:07 AM

Ahh darn, looks like I made an error right at the start. Thanks!

Also, you can stop after the first application of L'Hopital, since you can divide by the largest powers to get 3/2. (that's a shortcut we learnt at uni)

**undefined** · September 9th 2010, 01:17 AM

Originally Posted by Glitch

Ahh darn, looks like I made an error right at the start. Thanks!

Also, you can stop after the first application of L'Hopital, since you can divide by the largest powers to get 3/2. (that's a shortcut we learnt at uni)

is 3/2 a typo? denominator has higher degree polynomial so limit is 0.

**Glitch** · September 9th 2010, 01:27 AM

Prove It had the right answer (3/2), then he changed it for some reason.

**Glitch** · September 9th 2010, 01:32 AM

$\lim_{x \to \infty}\frac{\ln{(x^3 + 1)}}{\ln{(x^2 + 1)}} = \lim_{x \to \infty}\frac{\frac{3x^2}{x^3 + 1}}{\frac{2x}{x^2 + 1}} = \frac{3x^2}{x^3 + 1}.\frac{x^2 + 1}{2x} = \frac{3x^2(x^2 + 1)}{2x(x^3 + 1)} = \frac{3x(x^2 + 1)}{2(x^3 + 1)} = \frac{3x^3 + 3x}{2x^3 + 2} = \frac{3}{2}$

Unless I'm mistaken. (the textbook states 3/2 as the answer)

**Prove It** · September 9th 2010, 02:00 AM

Gah hang on, I'll fix it. It's well known my algebra is shocking...

**Glitch** · September 9th 2010, 02:03 AM

I assure you mine is worse. :P

**undefined** · September 9th 2010, 04:51 AM

Originally Posted by Glitch

Prove It had the right answer (3/2), then he changed it for some reason.

Ah, didn't bother checking the algebra, just saw that at the end of the first application of L'Hopital's rule Prove It had a cubic over a quartic (now corrected).

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