Without using L'Hopital's rule, how can I calaculate this limit?

[Calabi_Yau]

Without using L'Hopital's rule how can I calculate the limit of this function: (xⁿ-aⁿ)/(x-a) when x→a

I cannot get rid of the indeterminations no matter what. I would like if you could help me out on this.

 

[Dick]

Without using L'Hopital's rule how can I calculate the limit of this function: (xⁿ-aⁿ)/(x-a) when x→a

I cannot get rid of the indeterminations no matter what. I would like if you could help me out on this.

Factor x^n-a^n. For example x^2-a^2=(x-a)(x+a). x^3-a^3=(x-a)(x^2+xa+a^2) etc.

 

[Ray Vickson]

Without using L'Hopital's rule how can I calculate the limit of this function: (xⁿ-aⁿ)/(x-a) when x→a

I cannot get rid of the indeterminations no matter what. I would like if you could help me out on this.

Is n a positive integer? If so, just factor, as Dick has already suggested. If n is not a positive integer, you more-or-less need to use l'Hosptial's rule, whether you want to or not.

 

[Calabi_Yau]

I figured that out. Took me a while to notice if I factorized x^n-a^n beginning with (x-a)(...) the sum of the other factors would add up to n.a^n-1.

PS: I apologize not having posted my original atempt in solving the problem like the rules require.