Prove that lim n→∞ (1+1/n)^mn=e^m if m is an integer

 If m is an integer then prove that lim  n→∞      (1+1/n)^{^mn}=e^{^m}.  We will show you how you can prove lim  n→∞      (1+1/n)^{^mn}=e^{^m} when m is any integer.

Steps to prove that lim  n→∞      (1+1/n)^{^mn}=e^{^m}

Step1: lim  (n→∞ )   (1+1/n)^{^mn}=e^{^m}

                    

Step2: Use this formula, 

lim  (n→∞ ) [(1+c/n)ⁿ= e^{^c}]

                                                    

 

Step3:   lim  (n→∞ )   [(1+1/n)^{^n}]^{^m}

                 

∴ (1+1/n)^{^n}=e

Thus,  lim n→∞ [(1+1/n)^{^n}]^{^m}= e^{^m}

Hence, it is proved that [(1+1/n)^{^n}]^{^m}= e^{^m}, when m is an integer.

See Also: limit of (1+2/n)^n || step-by-step Explanation