Basic Limit Question

Zarlucicil

This is a pretty basic limit question regarding the limit,

[tex] \lim_{x \rightarrow \infty} (1+\frac{1}{x})^x = e [/tex]

Wolframalpha gives the following reasoning for this answer:

[tex] \lim_{x \rightarrow \infty} (1+\frac{1}{x})^x = e^{\lim_{x \rightarrow \infty} x\ln{(1+\frac{1}{x})}} = e^{\lim_{t \rightarrow 0} \frac{\ln{(1+t)}}{t}} = e^{\lim_{t \rightarrow 0} \frac{ \frac{d\ln{(1+t)}}{dt}}{\frac{d}{dt}t}}= e^{\lim_{t \rightarrow 0} \frac{1}{1+t}} = e^1 [/tex]

My question is, by the same reasoning, why is the following not true? (where log is log base 10)

[tex] \lim_{x \rightarrow \infty} (1+\frac{1}{x})^x = 10^{\lim_{x \rightarrow \infty} x\log{(1+\frac{1}{x})}} = 10^{\lim_{t \rightarrow 0} \frac{\log{(1+t)}}{t}} = 10^{\lim_{t \rightarrow 0} \frac{ \frac{d\log{(1+t)}}{dt}}{\frac{d}{dt}t}}= 10^{\lim_{t \rightarrow 0} \frac{1}{1+t}} = 10^1 [/tex]

Am I missing something??

Zarlucicil

Wooooow, hold on, I understand what I did wrong...

The derivative of [tex] \log{(1+t)} [/tex] is NOT [tex] \frac{1}{1+t} [/tex] but rather [tex] \frac{1}{(1+t)\ln{10}} [/tex]

Sorry for wasting the time of those who've read this.