Limit of a function within a function

Question

If limit as x->3 of (f(x)-4)/(x-3)=3, find the limit as x->3 of f(x).

 

If limit as x->3 of (f(x)-4)/(x-3)=6, find the limit as x->3 of f(x).

Answer 1

if (f(x)-4)/(x-3)=3 when x -> 3, the denominator must drop out because otherwise the denominator is zero.

 

Therefore the numerator must have a factor (x - 3) as well. And the value of the numerator when x = 3 must be 3.

 

So f(x) - 4 must have a factor (x - 3) and (x + a)

 

And x + a - 4 = 3 when x = 3 which means a = 4

 

Therefore f(x) = (x - 3)(x + 4) = x² + x - 12

 

Now you can do the second question in a similar way.