How do I evaluate left and right limits?

Question

I have this assignment:

$$\lim_{x \to 1} \frac{x^2 - 1}{|1 - x^3|}$$

I do not understand how I should do to separate this into two problems (one for $x \to 1^-$ and one for $x \to 1^+$ and then get rid of the absolute value. How do I do that?

Answer 1

If $x \gt 1$, then $|1-x^3|=x^3-1$, whereas if $x \lt 1$, then $|1-x^3|=1-x^3$. You can always separate a limit into $\lim_{x \to 1^+}$, which means you are just considering values of $x$ that are greater than $1$ and $\lim_{x \to 1^-}$, considering values of $x$ that are less than $1$. It is not always useful, but when you have absolute value signs around it can be. To have a two-sided limit, both these have to exist and they have to agree. So you would write $$\frac {x^2-1}{|1-x^3|}=\begin {cases} \frac {x^2-1}{x^3-1}&x \gt 0 \\ \frac {x^2-1}{1-x^3} & -1 \lt x \lt 0 \end {cases}$$ and take the right side limit of the first, the left side limit of the second.