FIND THIS LIMIT HELP PLEAASEEE!

Question

limx^m-1/xⁿ-1 m,n elements of N

x→1 

 the answer is m/n but i have no idea how to start or solve this! 

Answer 1

Solution 1 (factoring):

 

Note that xⁿ - 1 = (x - 1)(x^n-1 + x^n-2 + ... + x + 1).

 

 

lim_x→1 (x^m - 1)/(xⁿ - 1)

 

= lim_x→1 (x^m-1 + x^m-2 + ... + x + 1)/(x^n-1 + x^n-2 + ... + x + 1)

 

=  (1^m-1 + 1^m-2 + ... + 1¹ + 1⁰)/(1^n-1 + 1^n-2 + ... + 1¹ + 1⁰)

 

= m/n

 

 

Solution 2 (L'Hopital's Rule):

 

The limit has the 0/0 form so.

 

lim_x→1 (x^m - 1)/(xⁿ - 1)

 

= lim _x→1 (mx^m-1)/(nx^n-1)

 

=  (m·1^m-1)/(n·1^n-1)

 

= m/n