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Expansions to Remove Indeterminate Form

Evaluate the ...

Question

Evaluate the limit:

$lim x \to 0 \frac{x c o s x + 2 s i n x}{x^{2} + t a n x}$

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Solution

We have,
$lim x \to 0 \frac{x c o s x + 2 s i n x}{x^{2} + t a n x}$

It becomes 0/0 form.

$= lim x \to 0 \frac{x (c o s x + \frac{2 s i n x}{x})}{x (x + \frac{t a n x}{x})}$

$= lim x \to 0 \frac{(c o s x + \frac{2 s i n x}{x})}{(x + \frac{t a n x}{x})}$

$[∵ lim x \to 0 \frac{s i n x}{x} = 1, lim x \to 0 \frac{t a n x}{x} = 1, lim x \to 0 c o s x = 1]$

$= \frac{1 + 2 \times 1}{0 + 1}$

$= 3$

Therefore,

$lim x \to 0 \frac{x c o s x + 2 s i n x}{x^{2} + t a n x} = 3$

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Expansions to Remove Indeterminate Form

Standard XII Mathematics

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