As the title suggest, i am currently working on an exercise which asks me to prove that $$\lim_{x \to 0} \log_{10}{|x|}$$
does not exist. The proof is via contradiction.
My approach so far has been this:
Proof. Denote the limit by $\mathbf{L} \in \mathbb{R}$. By definition of a limit, there must be a $\delta \in \mathbb{R}_{>0}$ such that $$0<|x-0|<\delta$$ implies $$|\log_{10}|x|-\mathbf{L}|<\varepsilon.$$ for all $\varepsilon \in \mathbb{R}_{>0}$.
Let $\varepsilon = \frac{1}{2}$. Suppose for the sake of contradiction that there is a $m$ large enough such that $10^{-m}<\delta$. Thus $$0<|10^{-m}|<\delta$$ implies $$|\log_{10}|10^{-m}|-\mathbf{L}|=|(-m\cdot\log_{10}|10|)-\mathbf{L}|=|-m-\mathbf{L}|<\frac{1}{2}. \quad \blacksquare$$
And that is pretty much where i am stuck. The value for $\varepsilon$ whas randomly choosen, as i haven't figured which value it needs to be to create the contradiction.
I know, beggars can't be choosers, but if possible i only want a hint to nudge me in the right direction, so that i can figure out the rest of the proof by myself.
Thanks in advance :)