How to transform this limit $\lim_{h \to 0}\left(1+h\right)^{\frac{x}{h}}=e^x$ into $\lim_{h \to 0}\left(1+xh\right)^{\frac{1}{h}}=e^x$?

Question

How to transform this limit $\lim_{h \to 0}\left(1+h\right)^{\frac{x}{h}}=e^x$ into $\lim_{h \to 0}\left(1+xh\right)^{\frac{1}{h}}=e^x$?

This seems like I can't simplify this limit because I end up with zero divisors.

Can anyone help me on this?

Answer 1

Just set $t=xh \Rightarrow \frac{1}{h} = \frac{x}{t}$.

Hence, $$\lim_{h\to 0} (1+xh)^{\frac{1}{h}}= \lim_{t\to 0}\left((1+t)^{\frac{1}{t}}\right)^x = e^x$$

Additional info after comment:

The other way round works a bit differently.

You start with $(1+h)^{\frac{1}{h}}$.

Now, you need to squeeze in the $x$. Naturally, in this case, you have $h=x\frac{h}{x}$. So, $t = \frac{h}{x}$.

This would finally lead to $\lim_{t\to 0}\left((1+xt)^{\frac{1}{t}}\right)^{\frac{1}{x}} = e$.

Raising this to the power of $x$ gives $\lim_{t\to 0}(1+xt)^{\frac{1}{t}} = e^x$

Answer 2

Since $\lim_{h\to0}(1+h)^{\frac1h}=e$, you have$$\lim_{h\to0}(1+xh)^{\frac1{xh}}=e$$too and therefore\begin{align}\lim_{h\to0}(1+xh)^{\frac1h}&=\lim_{h\to0}\left((1+xh)^{\frac1{xh}}\right)^x\\&=\left(\lim_{h\to0}(1+xh)^{\frac1{xh}}\right)^x\\&=e^x.\end{align}