Sum

Find k, if the function f is continuous at x = 0, where

`f(x)=[(e^x - 1)(sinx)]/x^2`, for x ≠ 0

= k , for x = 0

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#### Solution

Since f is continuous at x = 0.

`lim_( x -> 0 ) f(x) = f(0) `

Given f(0) = k

∴ `lim_( x -> 0) f(x) = k` (i)

Now `lim_( x -> 0) f(x) = lim_( x -> 0 ) [(e^x - 1)sinx]/[x^2]`

= `lim_( x -> 0) ([e^x - 1]/[x])([sin x]/[x])`

= `lim_( x -> 0) ([e^x - 1]/[x]) lim_(x->0)([sin x]/[x])`

= log e x 1

= 1 x 1

`therefore lim_( x -> 0 ) = f(x) = 1` (ii)

from (i) and (ii)

k = 1

Concept: Continuous Function of Point

Is there an error in this question or solution?

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