Evaluate the limit of $\frac{1}{\log(x)}$

Question

Evaluate the limit $$\lim_{x\to1}\frac{1}{\log(x)}$$

The graph shows that $$\lim_{x\to1^-}\frac{1}{\log(x)} = -\infty$$

and that $$\lim_{x\to1^+}\frac{1}{\log(x)}=\infty$$

How can I explain this algebraically? Since the right side dows not equal the left side the limit does not exist right? Because of that it cannot exist as an extended real number?

Answer 1

The limit doesn't exist because the left-hand limit does not equal the right-hand limit

$$-\infty=\lim_{x\to1^-}\frac{1}{\log(x)} \neq \lim_{x\to1^+}\frac{1}{\log(x)}=\infty$$

if we extended the real number line and added positive and negative infinity then the limit still wouldn't exist because $\infty \neq -\infty$.