Find value of k to make limit exist and be finite

Question

Lim_x→5 ( x² +kx-20)/(x-5) 

 

Hi, I would really detailed steps. I know how to do this type pf problem when its piecewise but not sure on how to solve when its a rational. 

Answer 1

The denominator's factor, x-5, must cancel out when dividing into the numerator.  Therefore the numerator must be (x-5)(x+4).  This tells you what k shall be.

Answer 2

With the limit having to be a finite value, the denominator (x - 5) must be canceled out from the numerator otherwise there would be division by zero as x->5, which would not be a finite limit

 

For the cancellation to occur x =5 must b a solution of the numerator, i.e. x²+5k-20 = 0

 

Put this value of 5 into the numerator, 5² + 5k - 20 = 25 + 5k - 20 = 5 + 5k = 0.

Then k = -1

 

Check

 

lim_x->5(x² - x - 20)/(x - 5) = lim_x->5(x - 5)(x + 4)/(x - 5)

                                        = lim_x->5(x + 4) = 9