Thread: Definition of e in limit

1. Definition of e in limit

How do I prove that:
$\lim h \to 0 \left ( 1+h+\frac{h^{2}}{2} \right )^{\frac{1}{h}}=e$

I know that the definition of e is:
$\lim n \to \infty \left ( 1+\frac{1}{n} \right )^{n} =e$
Do I compare these two limits to get the answer?

2. Re: Definition of e in limit

Originally Posted by bilalsaeedkhan
How do I prove that:
$\lim h \to 0 \left ( 1+h+\frac{h^{2}}{2} \right )^{\frac{1}{h}}=e$

I know that the definition of e is:
$\lim n \to \infty \left ( 1+\frac{1}{n} \right )^{n} =e$
Do I compare these two limits to get the answer?
$\displaystyle \lim_{h \to 0}\left(1 + h + \frac{h^2}{2}\right)^{\frac{1}{h}} &= \lim_{h \to 0}e^{\ln{\left(1 + h + \frac{h^2}{2}\right)^{\frac{1}{h}}}} \\ &= \lim_{h \to 0}e^{\frac{\ln{\left(1 + h + \frac{h^2}{2}\right)}}{h}} \\ &= e^{\lim_{h \to 0}\frac{\ln{\left(1 + h + \frac{h^2}{2}\right)}}{h}} \\ &= e^{\lim_{h \to 0}\frac{\frac{1 + h}{1 + h + \frac{h^2}{2}}}{1}} \textrm{ by L'Hospital's Rule} \\ &= e^{\lim_{h \to 0}\frac{1 + h}{1 + h + \frac{h^2}{2}}} \\ &= e^{\frac{1 + 0}{1 + 0 + \frac{0^2}{2}}} \\ &= e^1 \\ &= e$