"Sincelimn⇒∞(1+1n)n={e}" was answered by our experts

chingli20013a1

Answered question

2022-04-28

Since

lim_{n \Rightarrow \infty} {(1 + \frac{1}{n})}^{n} = {e}

Assainato9gf

Beginner2022-04-29Added 15 answers

Suppose

lim_{n \to \infty} {(1 + \frac{r}{n})}^{n} = L

Let us calculate

\ln (L)

\ln (L) = \ln (lim_{n \to \infty} {(1 + \frac{r}{n})}^{n})

= lim_{n \to \infty} \ln ({(1 + \frac{r}{n})}^{n})

= lim_{n \to \infty} n \ln (1 + \frac{r}{n})

= lim_{n \to \infty} n \ln (1 + \frac{r}{n}) = lim_{n \to \infty} \frac{\ln (1 + \frac{r}{n})}{\frac{1}{n}}

= lim_{n \to \infty} \frac{r}{1 + \frac{r}{n}}

= r,

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