("lim")(x -> 0)((1+tanx)/(1+sinx))^(cos ...

`("lim")_(x -> 0)((1+tanx)/(1+sinx))^(cos e cx)` is equal to

(a)e

(b) `1/e`

(c) 1

(d) none of these

A

`underset(xto0)limf(x)` exists for `ngt0`

B

`underset(xto0)limf(x)` does not exists for `nlt0`

C

`underset(xto0)limf(x)` does not exists for any value of n

D

`underset(xto0)limf(x)` exists for any value of n

Text Solution

Verified by Experts

The correct Answer is:

C

`underset(xto0)lim((1+tanx)/(1+sinx))^(cosecx)=underset(xto0)lim((1+tanx)^((1)/(sinx)))/((1+sinx)^((1)/(sinx)))`
`=(underset(xto0)lim((1+tanx)^((1)/(tanx)))^((1)/(cosx)))/((1+sinx)^((1)/(sinx)))`
`=(e^((1)/(cos0)))/(e)=1`

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