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प्रश्न
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
उत्तर
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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