1. ## Trigonometric limit problem

$\lim_{x \to a} \frac{sqrt[tanx] - sqrt[tana]}{\sqrt[3]{sinx} - \sqrt[3]{sina}}$

Can anybody help?

Thank you.

2. Any idea?

3. Originally Posted by GreenMile
$\lim_{x \to a} \frac{sqrt[tanx] - sqrt[tana]}{\sqrt[3]{sinx} - \sqrt[3]{sina}}$

Can anybody help?

Thank you.
Try multiplying by $\frac{x-a}{x-a}$ to get

$\lim_{x\to a} \frac{\sqrt{\tan{x}}-\sqrt{\tan{a}}}{x-a} \cdot \frac{x-a}{\sqrt[3]{\sin{x}} - \sqrt[3]{\sin{a}}}$

Notice that both can be evaluated using the definition of the derivative at a point:
$f'(a) = \lim_{x \to a} \frac{f(x)-f(a)}{x-a}$