Question

For x > 0, `lim_(x rightarrow 0) ((sin x)^(1//x) + (1/x)^sinx)` is ______.

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Answer 1

For x > 0, `lim_(x rightarrow 0) ((sin x)^(1//x) + (1/x)^sinx)` is 1.

Explanation:

Here, `lim_(x rightarrow 0) (sin x)^(1//x) + lim_(x rightarrow 0) (1/x)^sinx`

= `0  + lim_(x rightarrow 0) e^(log(1/x)^sinx)`  ...`[(∵ lim_(x rightarrow 0) (sin x)^(1//x)  rightarrow 0),("as"  0 < sin x < 1)]`

= `e^(lim_(x rightarrow 0)) (log(1/x))/("cosec"  x)`  ...[by L’ Hopsital rule]

= `e^(lim_(x rightarrow 0)) (x((-1)/2))/(-"cosec"  x cot x)`  

= `e^(lim_(x rightarrow 0)) sinx/x tan x`

= e⁰

= 1