Q:

Finding limit of the following function

I need to find $$\lim_{x\to a}(\log_ax)^{1/(x-a)}$$
Isnt the answer simply 1 as the logarithm inside is actually 1 and not tending to 1 to make it an indeterminant form.

A:

When you compute a limit, say
$$\lim_{x\to a}f(x),$$
all values of $x$ in a neighborhood of $a$ are considered. But not $x=a$. Hence you can't have $\log_ax=1$.

By the way, if you assume that $\log_ax=1$, for coherence you must also admit that $x-a=0$ and the expression under the limit would be $1^{1/0}$, which does not make sense.
Of course, it is true that
$$\lim_{x\to a}(\log_aa)^{1/(x-a)}=1$$
but this is a quite different question.

