lim as x approaches infinity (1/(k^3 +1))^1/k = 1 but why?

Question

lim as x approaches infinity (1/(k^3 +1))^1/k  = 1 but why?

anonymous · Answer

One method you can do is take the natural log and see if that approaches anything.

anonymous · Answer

$$
\lim_{k\to \infty}\left(\frac 1{k^3+1}\right)^{1/k}
$$

anonymous · Answer

Considering how $x = e^{\ln(x)}$ when $x>0$, we can take the $\ln$ and see what happens to the expression: $$
\frac 1k \bigg( \ln(1) - \ln(k^3+1)\bigg)  = -\frac{\ln(k^3+1)}{k}
$$

anonymous · Answer

Now, typically it is understood that $x$ dominates $\ln x$, so this will converge to $0$.

$e^{\ln x} = e^0 = 1$

anonymous · Answer

oh, ok that makes a lot of sense, thanks

anonymous · Answer

As k tends to infinity direct sustituion of limits will gt u 0^0 which is an undefined form inmath

anonymous · Answer

@wio Follow this method works fine for some case for more complicated cases there r some theorems u can use to handle these undefined forms