how is this happening?

**TYTY** · March 25th 2009, 08:06 AM

$<br /> \frac{\ln{\frac{x+3}{x+1}}}{1/x} = \frac{2x^2}{(x + 1)(x + 3)}<br />$

This is related to trying to get lim->infinity so technically it is calculus

**Soroban** · March 25th 2009, 08:32 AM

Hello, TYTY!

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

The limit goes to $\tfrac{0}{0}$ . . . so we can apply L'Hopital

We have: . $\frac{\ln (x+3) - \ln(x+1)}{x^{-1}}$

Apply L'Hopital: . $\frac{\frac{1}{x+3}-\frac{1}{x+1}} {-x^{-2}} \;=\;\frac{\frac{-2}{(x+1)(x+3)}}{-\frac{1}{x^2}} \;=\;\frac{2x^2}{(x+1)(x+3)}$

**TYTY** · March 25th 2009, 08:39 AM

Originally Posted by Soroban

Hello, TYTY!

The limit goes to $\tfrac{0}{0}$ . . . so we can apply L'Hopital

We have: . $\frac{\ln (x+3) - \ln(x+1)}{x^{-1}}$

Apply L'Hopital: . $\frac{\frac{1}{x+3}-\frac{1}{x+1}} {-x^{-2}} \;=\;\frac{\frac{-2}{(x+1)(x+3)}}{-\frac{1}{x^2}} \;=\;\frac{2x^2}{(x+1)(x+3)}$

Good lord I think my brain just exploded.

you rock soroban

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