# How do you find the limit of (1+7/x)^(x/10) as x approaches infinity?

Apr 28, 2016

There are other method available, but I use ${\lim}_{m \rightarrow \infty} {\left(1 + \frac{1}{m}\right)}^{m} = e$

#### Explanation:

${\left(1 + \frac{7}{x}\right)}^{\frac{x}{10}} = {\left({\left(1 + \frac{7}{x}\right)}^{x}\right)}^{\frac{1}{10}} = {\left[{\left(1 + \frac{1}{\frac{x}{7}}\right)}^{\frac{x}{7}}\right]}^{\frac{7}{10}}$

As $x \rightarrow \infty$ the quotient $\frac{x}{7} \rightarrow \infty$ as well.

So, with $m = \frac{x}{7}$ we see that the expression in the brackets goes to $e$. That is

${\lim}_{m \rightarrow \infty} \left[{\left(1 + \frac{1}{\frac{x}{7}}\right)}^{\frac{x}{7}}\right] = e$

Therefore,

${\lim}_{x \rightarrow \infty} {\left(1 + \frac{7}{x}\right)}^{\frac{x}{10}} = {\lim}_{x \rightarrow \infty} {\left[{\left(1 + \frac{1}{\frac{x}{7}}\right)}^{\frac{x}{7}}\right]}^{\frac{7}{10}}$

$= {\left[{\lim}_{x \rightarrow \infty} {\left(1 + \frac{1}{\frac{x}{7}}\right)}^{\frac{x}{7}}\right]}^{\frac{7}{10}}$

$= {e}^{\frac{7}{10}}$