For what function f(x) and number a is the the limit `lim_(x->4) (2^x-16)/(x-4)` equal to the value of f'(a)?

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justaguide | College Teacher | (Level 2) Distinguished Educator

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The value of `f'(a) = lim_(x->4)(2^x - 16)/(x - 4)`

Substituting x = 4 gives an indeterminate form `0/0` . This allows l'Hopital's rule to be used and the numerator and denominator are substituted by their derivatives.

`lim_(x->4) (ln 2*2^x)/1`

Substituting x = 4 gives `ln 2*16`

There can be many combinations of functions f(x) and values of a for which f'(a) = `ln 2*16` , one example is `f(x) = ln 2*x^2/2`

`f'(x) = ln 2*x`

`f'(16) = ln 2*16`

This gives one solution as `f(x) = ln 2*(x^2/2)` and `a = 16`

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