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Question
\[\lim_{x \to 0} \frac{7x \cos x - 3 \sin x}{4x + \tan x}\]
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Solution
\[\lim_{x \to 0} \left[ \frac{7x \cos x - 3 \sin x}{4x + \tan x} \right]\] It is of the form \[\left( \frac{0}{0} \right)\]
Dividing the numerator and the denominator by x:
\[\Rightarrow \lim_{x \to 0} \frac{7\cos x - 3\left( \frac{\sin x}{x} \right)}{4 + \left( \frac{\tan x}{x} \right)}\]
\[ \Rightarrow \frac{7 \lim_{x \to 0} \left( \cos x \right) - 3 \lim_{x \to 0} \left( \frac{\sin x}{x} \right)}{4 + \lim_{x \to 0} \left( \frac{\tan x}{x} \right)}\]
\[ \Rightarrow \frac{7 . 1 - 3 . 1}{4 + 1}\]
\[ \Rightarrow \frac{4}{5}\]
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