• Contact
• About

  Search by keyword:  

Last modified: January 26, 2018

The Limit Definition of the Exponential Function
-----------------------------------------------

exp(x) = lim (1 + x/n)n
         n->∞

Proof:

Let f(x) = ln(x)

f'(x) = 1/x 

      = lim {ln(x + h) - ln(x)}/h
        h->0 
 
      = lim (1/h)ln((x + h)/x)
        h->0   

      = lim ln((x + h)/x)1/h
        h->0   

We can write:

exp(1/x) = exp{lim ln((x + h)/x)1/h}
               h->0

         = lim exp{ln((x + h)/x)1/h}
           h->0

         = lim (x + h)/x)1/h
           h->0

         = lim (1 + h/x)1/h
           h->0

Let R = 1/x and n = 1/h

exp(R) = lim (1 + R/n)n    Q.E.D.
         n->∞

 

Copyright © 2023 Redshift Academy

This website built by me using the Codeigniter MVC framework