Given `f(x)=sqrt(5x-1) - 3` , find `lim_(h->0)(f(2+h)-f(2))/h`

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justaguide's profile pic

justaguide | College Teacher | (Level 2) Distinguished Educator

Posted on

The function `f(x) = sqrt(5x - 1) - 3`

`lim_(h->0) (f(2+h) - f(2))/h`

=> `lim_(h->0) (sqrt(5*(2+h) - 1) - 3 - sqrt(5*2 - 1) + 3)/h`

=> `lim_(h->0) (sqrt(10 + 5h - 1) - sqrt 9)/h`

=> `lim_(h->0) ((sqrt(9 + 5h) - sqrt 9)(sqrt(9 + 5h) + sqrt 9))/(h*(sqrt(9 + 5h) + sqrt 9)`

=> `lim_(h->0) (5h)/(h*(sqrt(9 + 5h) + sqrt 9)`

=> `lim_(h->0) 5/(sqrt(9 + 5h) + sqrt 9)`

substituting h = 0

=> `5/6`

The value of `lim_(h->0) (f(2+h) - f(2))/h` for `f(x) = sqrt(5x - 1) - 3` is `5/6`

Wiggin42's profile pic

Wiggin42 | Student, Undergraduate | (Level 2) Valedictorian

Posted on

Given  , find 

The limit given is in fact the formal definition of a derivative. You are being asked to find the derivative of f(x) at the point x = 2. 

so f(x) = `(5x - 1)^(1/2) - 3`

`f'(x) = (5)/(2sqrt(5x - 1))`

`f'(2) = 5/6`

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