Question

Is f(x) =(1+x)^1/x,when x is not equal to 0 and f(x)=e, when x=0 continuous at x=0.

Answer 1

By definition:    Euler's exponential function is defined as:
          eⁿ =  Lim x -> 0,  (1 + x n)^(1/x)    or,
                                      (1 + n / x )^x

Given    f(x) = (1 + x)^(1/x)   when x ≠ 0
 So  Lim x -> 0 ,  f(x) = Lim x -> 0,  (1 + 1* x)^(1/x)
                                  = e¹ = e.
Also given  f(0) = e.

Since the limit as x tends to 0 is same as f(0) , the function f(x) is continuous at x = 0.
==========================