Mathematics: Limit and Continuity

Limits

A function f(x) is said to have a limit at l at point a when f(x) approaches to l whenever x approaches to a.

(lim)┬(x→a)⁡f(x)=l

Important Theorem on Limit

For all rational values of n,   (lim)┬(x→a)⁡〖(x^n-a^n)/(x-a)〗=na^(n-1)  

Case 1:

When n is a positive integer:

By actual division,  (x^n-a^n)/(x-a)= x^(n-1)+x^(n-2).a+x^(n-3).a^2+…+a^(n-1)

Now,

(lim)┬(x→a)⁡〖(x^n-a^n)/(x-1)〗  =   (lim)┬(x→a)⁡[x^(n-1)+x^(n-2).a+x^(n-3).a^2+…+a^(n-1) ]

                = a^(n-1)+a^(n-1)+a^(n-1)+…+a^(n-1)
                = na^(n-1)