(1+x)^(1/z) - Limit as x approaches 0

[seasponges]

I was given this as a practice question, and assumed that the answer would be 0.

The answer is e, but no explanation is given and I cannot figure out why this is. z is undefined in the question.

If anyone could shed some light it would be greatly appreciated.

 

[Curious3141]

I was given this as a practice question, and assumed that the answer would be 0.

The answer is e, but no explanation is given and I cannot figure out why this is. z is undefined in the question.

If anyone could shed some light it would be greatly appreciated.

There shouldn't be a z, it should simply be replaced by x.

$$\lim_{x \rightarrow 0} {(1 + x)}^{\frac{1}{x}} = e$$

That's actually one of the definitions for e (the base of natural logarithms). If you want a fairly elementary (but not very rigorous) way of proving it, consider putting $y = \frac{1}{x}$, from which you get the equivalent definition:

$$\lim_{y \rightarrow \infty} {(1 + \frac{1}{y})}^y = e$$

and apply Binomial theorem to the LHS. Expand and consider the limit as y tends to infinity. Now compare that with the Taylor series for e (e¹).

 

[seasponges]

Thankyou very much!