2
$\begingroup$

I am trying to find $\lim_{x \rightarrow0} (1-3 \cdot x)^{\frac{1}{x}}$

I thought about finding the limit of

$$(1-3 \cdot x)^{\frac{1}{x}}= e^{\frac{\ln(1-3 \cdot x)}{x}}$$

But that only works if $e^{\frac{\ln(1-3 \cdot x)}{x}}$ is continuous at $x=0$, which as far as I understand it is not. Am I right about that? And if I am, how do I find the limit?

Thanks!

  • 0
    Continuity **at** $0$ is irrelevant. To use many arguments, we do want the function to be continuous in a *deleted neighbourhood* of $0$. So there should exist an $\epsilon$ such that the function is continuous at all $x$ such that $0<|x|<\epsilon$.2012-05-21

4 Answers 4