Lim x-->∞ (1+1/x)x
The limit lim x→∞ (1+1/x)x is of the indeterminate form 1∞. Before we can use l'Hospital's Rule, we need to bring it into the standard indeterminate form 0/0. For this we use the fact that
lim (ln f(x)) = ln (lim f(x))
y = ln (1+1/x)x = x ln (1+1/x) = ln(1+1/x) / (1/x)
limx→∞ y = limx→∞ ln(1+1/x) / (1/x)
which is of the form 0/0, so by l'Hospital's Rule,
limx→∞ y = lim (1/(1+1/x))(-1/x²) / (-1/x²) = lim 1/(1+1/x) = 1.
ln (lim x→∞ (1+1/x)x) = 1
lim x→∞ (1+1/x)x = e.
Note: this is how you get the continuous compound interest formula!