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प्रश्न
Prove that the function f given by `f(x) = |x - 1|, x in R` is not differentiable at x = 1.
उत्तर
Any function will not be differentiable if the left hand limit and the right hand limit are not equal.
f(x) = `abs (x - 1), x in R`
f(x) = (x - 1), if x - 1 > 0
= - (x - 1), if x -1 < 0
At x = 1
f(1) = 1 - 1 = 0
left side limit =
`lim_(h -> 0^-) (f(1 - h) - f(1))/ -h`
= `lim_(h -> 0^-) (1 - (1 - h) - 0)/ (- h)`
= `lim_(h -> 0^-) (+ h)/(- h)`
= - 1
Right side limit =
= `lim_(h -> 0^+) (f(1 + h) - f(1))/h`
= `lim_(h -> 0^+) ((1 + h) - 1 - 0)/ h`
= `lim_(h -> 0^+) h/h`
= 1
Left side limit and right side limit are not equal.
Hence, f(x) is not differentiable at x = 1.
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