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Question
For x > 0, `lim_(x rightarrow 0) ((sin x)^(1//x) + (1/x)^sinx)` is ______.
Options
0
1
–1
2
MCQ
Fill in the Blanks
Solution
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
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Continuous and Discontinuous Functions
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