Evaluate lim as h approaches 0 (3(x+h)^47 -(3X^47))/h

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justaguide's profile pic

justaguide | College Teacher | (Level 2) Distinguished Educator

Posted on

The limit `lim_(h->0) (3(x+h)^47 -(3x^47))/h` has to be determined.

If h is equated to 0 the form `0/0` is obtained which is indeterminate. This allows the use of l'Hopital's rule and the numerator and denominator are substituted by their derivatives.

Here as h tends to 0, the derivative is with respect to h

=> `lim_(h->0) (3*47*(x+h)^46 - 0)/1`

substitute h = 0

=> 141*x^46

The required limit is 141*x^46

Wiggin42's profile pic

Wiggin42 | Student, Undergraduate | (Level 2) Valedictorian

Posted on

Evaluate lim as h approaches 0 (3(x+h)^47 -(3X^47))/h

`lim_(h->0) ( 3(x+h)^47 - (3x^47) )/ (h)`

``There is a much simpler way to solve this than using L'Hopital's rule. It requires you to realize that this limit is in the form: 

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

This is the formal definition of the derivative. All you have to do is take the derivative of f(x)! 

In your case, f(x) = 3x^47. 

Therefore f'(x) = 141x^46 

and that is what the limit is equal to. 

` `

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