Limits of trivial expressions involving the logarithmic function

Question

By simplifying the trivial expression inside the limit, $$\lim_{x\to1}\frac{\frac{x}{x-1}}{\frac{x}{x-1}}=1$$ My question is: can we do the same (I suspect we can't) if $$\lim_{x\to1}\frac{\log\frac{x}{x-1}}{\log\frac{x}{x-1}}?$$

Answer 1

The expression inside the logarithm is identically $1$ when $x \ne 1$, and those are the only values of $x$ that matter when calculating the limit at $1$. So the answer is the limit of $\log (1)$, which is of course $0$.

The second expression is similar. When $x \ne 1$ the expression inside the limit is just $1/1$ so the limit is $1$.

Neither question really has much to do with logarithms, except that in the second question the logarithms are undefined when $x$ is between $0$ and $1$.