Maharashtra State BoardHSC Commerce 12th Board Exam
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Find K, If the Function F is Continuous at X = 0, Where F ( X ) = ( E X − 1 ) ( Sin X ) X 2 , for X ≠ 0 = K , for X = 0 - Mathematics and Statistics

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

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