Evaluate $\lim_{x \to 0}{\frac{\sqrt[3]{1 + cx} - 1}{x}}$ without using L-H Rule

Question

$$\lim_{x \to 0}{\frac{\sqrt[3]{1 + cx} - 1}{x}}$$ where $c$ is a constant.

Since given limit has $0/0$ form , So one can easily use L-H Rule and get ans as $c/3$ but How we find limit without using L-H Rule?

Answer 1

Hint: Multiply the numerator and denominator by $\sqrt[3]{1+cx}^2+\sqrt[3]{1+cx}+1$.

This is based on the idea that $(a-b)(a^2+ab+b^2)=a^3-b^3$. We use it with $a=\sqrt[3]{1+cx}$ and $b=1$, transforming the numerator into $cx$.

Answer 2

You can use the derivative number of the function $ \sqrt[3]{1+cx}$