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dks2114 · Jul 25, 2007, 08:36 AM

What is the lim x->0 (1+3x) ^ (1/x)

ebaines · Jul 25, 2007, 09:47 AM

You can solve this using the definition of e: this might help you:

Exponentiation - Wikipedia, the free encyclopedia

Think of n as used on this site as 1/x in your problem (i.e, this site talks about the limit as n approaches infinity, whereas you want the limit as x approaches 0). You'll see that your problem reduces to a power of e. Can you take it from here?

dks2114 · Jul 25, 2007, 05:14 PM

Can you use L'Hopital's method? Is that a legal move?

galactus · Jul 25, 2007, 05:46 PM

No, L'Hopital is probably not the best to tackle this one.

Use the fact that $\lim_{x\to\0}(1+x)^{\frac{1}{x}}=e$

Let t=3x and x=t/3.

$\lim_{t\to\0}(1+t)^{\frac{3}{t}}$

$\lim_{t\to\0}[\underbrace{(1+t)^{\frac{1}{t}}}_{\text{this is e}}]^{3}$

Now, see what you get as the limit?

dks2114 · Jul 26, 2007, 02:55 AM

Oh wow that's so much simpler. I used L'Hopital... still got the same answer but wow nty version was so much more complicated. Thank you very much!