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प्रश्न
Find the derivatives from the left and from the right at x = 1 (if they exist) of the following functions. Are the functions differentiable at x = 1?
`f(x) = |x - 1|`
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उत्तर
`f(x) = {{:(x - 1, "if" x > 1),(-(x - 1), "if" x < 1):}`
First we find left limit of `f(x)` at x = 1
When `x -> 1-` we have `f(x) = - (x - 1)`
`f"'"(1^-) = lim_(x -> 1^-) (f(x) - f(1))/(x - 1)`
`f"'"(1^-) = lim_(x -> 1) (-(x - 1) - (0))/(x - 1)`
= `lim_(x -> 1) (-(x - 1))/(x - 1)` = – 1 .......(1)
`f"'"(1^+) = lim_(x -> 1^+) (f(x) - f(1))/(x - 1)`
= `lim_(x -> 1^+) ((x - 1) - 0)/(x - 1)`
`f"'"(1^+) = lim_(x -> 1^+) (x - 1)/(x - 1)` = 1 ......(2)
From equation (1) and (2) we have
`lim_(x -> 1^-) f(x) ≠ lim_(x -> 1^+) f(x)`
∴ `f"'"(x)` does not exist at x = 1
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