Question

If f(x) is differentiable at x = 1 and `lim_(h → 0) 1/h f(1 + h) = 5`, then f' (1) is equal to ______.

विकल्प

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Answer 1

If f(x) is differentiable at x = 1 and `lim_(h → 0) 1/h f(1 + h) = 5`, then f' (1) is equal to 5.

Explanation:

`f'(1) = lim_(h → 0) (f(1 + h) - f(1))/h`

 = `lim_(h → 0) (f(1 + h))/h - lim_(h → 0) (f(1))/h`

Given, `lim_(h → 0) (f(1 + h))/h = 5`

So, `lim_(h → 0) (f(1))/h`, must be finite as f' (1) exists and

`lim_(h → 0) (f(1))/h` can be finite only, if f(1) = 0 and

`lim_(h → 0) (f(1))/h = 0`

So, `f'(1) = lim_(h → 0) (f(1 + h))/h = 5`