TITLE: What is $\mathop {\lim }\limits_{x \to 0} \frac{{\tan 2x + \tan 4x - \tan 6x}}{{{x^3}}}$? .
QUESTION [5 upvotes]: What is $\mathop {\lim }\limits_{x \to 0} \frac{{\tan 2x + \tan 4x - \tan 6x}}{{{x^3}}}$?

REPLY [2 votes]: $$\underset{x\to 0}{\mathop{\lim }}\,\frac{(\tan 2x-2x)+(\tan 4x-4x)+(6x-\tan 6x)}{{{x}^{3}}}=\frac{1}{3}\underset{x\to 0}{\mathop{\lim }}\,\frac{8{{x}^{3}}+64{{x}^{3}}-216{{x}^{3}}}{{{x}^{3}}}=-48$$