# Math Help - another exponential limit

1. ## another exponential limit

Thanks so much! The limit as n--> infinity of [1 + 7/n + 1/(n^4)]^n

2. Hello,

Originally Posted by winterwyrm
Thanks so much! The limit as n--> infinity of [1 + 7/n + 1/(n^4)]^n
$(1+\frac 7n+\frac 1{n^4})^n$

Taking the logarithm :

$n \ln(1+\frac 7n+\frac 1{n^4})$

We know that $\lim_{x \to 0} \ln(1+x)=\lim_{x \to 0} x$

Here, $\lim_{n \to \infty} \frac 7n+\frac 1{n^4}=0$.

So $\lim_{n \to \infty} n \ln(1+\frac 7n+\frac 1{n^4})=\lim_{n \to \infty} n( \frac 7n+\frac 1{n^4})=\lim_{n \to \infty} 7+\frac 1{n^3}=7$

By taking back the exponential (because we took the logarithm) :

$\lim_{n \to \infty}(1+\frac 7n+\frac 1{n^4})^n=\lim_{n \to \infty} e^{n \ln(1+\frac 7n+\frac 1{n^4})}=e^7$

3. Brighter than a thousand suns.