Proof of e as a limit

Question

I'm reading this text:

A few questions:

1. What's the importance of them going from $h$ to $x$ in the first line? What is the difference?

2. How did they go from $$\lim_{x \to 0} \frac{\ln(1+x) - \ln(1)}{x}$$ to $$\lim_{x \to 0} \left[\frac{1}{x} \cdot \ln(1+x)\right]$$

3. And then right before the blue 5 box... how did they go from: $$e^{\lim_{x \to 0} \ln(1+x)^{1/x}}$$ to $$\lim_{x \to 0} e^{\ln(1+x)^{1/x}}$$ How did they just pull out the limit sign?

Answer 1

The x-h notations are equivalent.

Note that $\ln 1=0$ thus

$$\lim_{x \to 0} \frac{\ln(1+x) - \ln(1)}{x}=\lim_{x \to 0} \left(\frac{1}{x} \cdot \ln(1+x)\right)$$

Since the exponential function is continuos $$e^{\lim f(x)}\equiv \lim e^{f(x)}$$

Answer 2

1. There is no difference.

2. $\ln(1)=0$.

3. The exponential function is continuous.