Question

If $\underset{x\to a}{\lim }\frac{x^3-a^3}{x-a}=\underset{x\to 1}{\lim }\frac{x^4-1}{x-1},$ find all possible values of a.

Answer 1

$$\begin{gathered} \underset{x\to a}{\lim }\left[\frac{x^3-a^3}{x-a}\right]=\underset{x\to 1}{\lim }\left[\frac{x^4-1^4}{x-1}\right] \\ \Rightarrow 3a^{3-1}=4{\left(1\right)}^{4-1} \\ \Rightarrow 3a^2=4 \\ \Rightarrow a^2=\frac{4}{3} \\ \Rightarrow a=\pm \frac{2}{\sqrt{3}} \end{gathered}$$