Find the constant C

Question

lim x→2 x^2+x+C/ x^2-3x+2 so that the limit exists 

Answer 1

In order for the limit to exist, the denominator cannot be zero, and there could be no indeterminate (0/0).  Let see what happens if we factor out the denominator.

 

(x - 2)(x - 1)

 

Plugging in x=2 into the denominator makes (x - 2) equal to zero.  WE DON'T WANT THAT.

 

So we do something to the numerator so that when we factor it, we can cancel out (x - 2).

 

x² + x + C = (x - 2)(x + 3) = (x² + x - 6)

 

 

So when C=-6,  the limit reduces to

 

(x - 2)(x + 3) / (x - 2)(x - 1) = (x + 3) / (x - 1)

 

 

Now the limit exists when x=2.  Therefore,    C = -6