Limit evaluation using algebra of sequences and sequence theorems

Question

By making use of only the theorems on sequences (ex: algebra of sequences/cauchy's first theorem of sequences/limit of geometric mean of a sequence etc), how to prove the following: $lim_{n\to\infty}(1+\frac{1}{n})^n = e$

Answer 1

$$\left( 1 + \frac{1}{n} \right)^n = e^{n \ln \left(1 + \frac{1}{n} \right)}$$

Now $$n \ln \left(1 + \frac{1}{n} \right) \sim n \times \frac{1}{n} \rightarrow 1$$

Answer 2

Simply expand it using binomial theorem cancel out the n's and put n= infinity now all the terms that will be remaining will contain expression like (1-1/n) and so on the term 1/n becomes zero as 1/infinity is zero. Now the resulting expression is the expansion of 'e' thus you get it as e