Important Limits

There are some important Limits :

1. Lt (x  →  a ) (xⁿ - aⁿ) / (x - a) = n*a^n-1 [ n is rational ] 

[ Prove is easy for n is integer expand the numerator and get the result by eliminating Limit operations . For other rational make n = p/q where p is integer and q is natural ]

2. Lt (x  →  0 ) sin(x)/x = 1 

3. Lt (x  →  0 ) ((1+x)ⁿ - 1)/x = n [ n is rational ]

4.  Lt (x  →  0 ) (e^x- 1)/x = 1 [ n is rational ]

Col.  Lt (x  →  0 ) (1+x)^1/x = e [ From above ]