Lim x - 0atan x-asin x-tan x-sin x- is equal to a-0

The correct option is B We have x→ 0lima^tan x a^sin x/tan x sin x=x→ 0lim a^sin x(a^tan x sin x 1/tan x sin x) x→ 0lim(a^sin x)×t→ 0lim ( a^t 1/t ) (where ...

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Byju's Answer

Standard XII

Mathematics

Integration by Partial Fractions

lim x → 0atan...

Question

$l i m x \to 0 \frac{a^{t a n x} - a^{s i n x}}{t a n x - s i n x}$ is equal to (a>0)

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Solution

The correct option is A
We have $l i m x \to 0 \frac{a^{t a n x} - a^{s i n x}}{t a n x - s i n x} = l i m x \to 0 a^{s i n x} (\frac{a^{t a n x - s i n x} - 1}{t a n x - s i n x}) l i m x \to 0 (a^{s i n x}) \times l i m t \to 0 (\frac{a^{t} - 1}{t})$ (where t=tan x-sinx)
$= a^{0} \times l o g_{e} a = l o g_{e} a$

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$l i m x \to 0 \frac{a^{t a n x} - a^{s i n x}}{t a n x - s i n x}$ is equal to (a>0)