Limit question

Question

Consider the limit,  lim(??x→1 )  ( x² −kx+k−1)/(x-1), where k is a real number,?? Compute the limit

 

The answer is 2-k, but do I get that?

 

thankyou for the help

Answer 1

The limit has the form 0/0, so we could use L'Hopital's Rule.  The limit can also be evaluated algebraically as follows:

 

lim_x→1[(x²-kx+k-1)/(x-1)]

 

= lim_x→1[((x²-1) + (k-kx))/(x-1)]

 

= lim_x→1[((x+1)(x-1) -k(x-1))/(x-1)]

 

= lim_x→1[((x-1)(x+1-k))/(x-1)]

 

= lim_x→1(x+1-k) = 1+1-k = 2-k

Answer 2

If you factor the numerator, you should get

 

(x -1)(x - (k-1))

 

and then you can cancel and evaluate the limit.