Can i ask for the limit as $x$ approaches $\lim_{x \to 0-}\ln x$? Please explain. Its because since the limit of a function only exists if the lim as $x$ approaches some number $n$ from both the positive and negative side is the same, im not sure that im convinced that the limit as $x$ approaches $0$ for $\ln x$ exists. i know its negative infinity from the positive side, but from the negative side?
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$\begingroup$The domain of $\ln(x)$ is only positive reals, so the left-hand limit at 0 doesn't really make sense.
$\endgroup$ 3 $\begingroup$For a limit to exist, the two limits approaching from the left and right side need to match up.
I.e. we need $$\lim\limits_{x\to0^-}\ln(x) = \lim\limits_{x\to0^+}\ln(x)$$ to be true.
Since $\ln(x)$ is not defined for $x\leq 0$ assuming we are evaluating over the reals, the left-hand limit can't be evaluated, and thus the limit does not exist.
If you are evaluating over the complex numbers, that's a somewhat different story, but given your wording, I've assumed that we're talking about the reals here.
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