Question

evaluate the limit

Answer 1

\[\lim_{n \rightarrow \infty} {(1+\frac{1}{n})(1+\frac{2}{n})(1+\frac{3}{n})....(1+\frac{n}{n})}^\frac{1}{n}\]

Answer 2

Limit properties state that \[\lim_{x\to a} f(x)g(x) = \lim_{x \to a} f(x) \lim_{x \to a} g(x) \]
Then \[ \lim_{n \to \infty} \left (1+\frac{a}{n} \right ) = 1 \] So essentially, every limit, but \[ \lim_{n \to \infty }\left ( 1+\frac{n}{n} \right )^{\frac{1}{n}}\] is 1. We just need to solve the limit above.