lim x->c- (c^2-x^2)/(|x-c|) e. Evaluate the limits or explain why it does not exist

Question

lim        (c^2-x^2)/(|x-c|)

x->c-

e. Evaluate the limits or explain why it does not exist

Answer 1

factor the numerator: (c-x)(c+x).

represent x approaching c from the left by setting x = c-ε where ε is a very small positive quantity.

 

then during the approach to c^- we can represent x as c-ε so that the factor (c-x) equals c-(c-ε) = ε.

In the denominator, the absolute value also becomes ε , and so these two factors cancel exactly.

 

Then we are simply taking the limit of (x+c) and this becomes the final answer, 2c.