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Question Number 117422 by bemath last updated on 11/Oct/20

$$\underset{x\to 0}{\lim }\frac{3\tan 4\mathrm{x}-4\tan 3\mathrm{x}}{3\sin 4\mathrm{x}-4\sin 3\mathrm{x}}=?$$

Answered by bobhans last updated on 11/Oct/20

$$\underset{x\to 0}{\lim }\frac{3\tan 4\mathrm{x}-4\tan 3\mathrm{x}}{3\sin 4\mathrm{x}-4\sin 3\mathrm{x}}=$$$$\underset{x\to 0}{\lim }\frac{3(4\mathrm{x}+\frac{1}{3}{\left(4\mathrm{x}\right)}^3)-4(3\mathrm{x}+\frac{1}{3}{\left(3\mathrm{x}\right)}^3)}{3(4\mathrm{x}-\frac{{\left(4\mathrm{x}\right)}^3}{6})-4(3\mathrm{x}-\frac{{\left(3\mathrm{x}\right)}^3}{6})}=$$$$\underset{x\to 0}{\lim }\frac{12\mathrm{x}+{64\mathrm{x}}^3-12\mathrm{x}-{36\mathrm{x}}^3}{12\mathrm{x}-{32\mathrm{x}}^3-12\mathrm{x}+{18\mathrm{x}}^3}=$$$$\underset{x\to 0}{\lim }\frac{{28\mathrm{x}}^3}{-{14\mathrm{x}}^3}=-2$$

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