Consider the limit lim f(x)g(x) x→a Spivak has proved that this is equal to lim f(x) multlied by x→a lim g(x) x→a And also if lim g(x) = k and k≠0, x→a Then. lim 1/g(x) = 1/k x→a Now the problem arises..... Consider the limit lim ((x^2)-(a^2))/(x-a) x→a It can factorised and written as( taking x-2 from numerator) lim (x+a) x→a Which is nothing but 2a. Now we can write it the above limit also as lim(x^2)-(a^2) multiplied by x→a lim 1/(x-a) x→a. The second limit does not exist because lim(x-a)=0 and l=0 x→a So, its reciprocal limit does not exist. Then can't we say lim ((x^2)-(a^2))/(x-a) does not exist? x→a Where am I wrong in my arguement?