Proof of limits with fraction

[joshua.sl]

prove that limit of --- (x$$^{n}$$-a$$^{n}$$)
---------- ---------------- (x-a)
when " x " tends to " a "

equals to ---- n.a$$^{n-1}$$ ----- ****where " n " is a fraction..


we have been given this and I've got no idea on how to prove it although i can do sums with this equation when " n " is a fraction


PS this is my first post and i have no experience in witing equations in posts ..sorry if I've done any mistakes.

 

[lippyka]

Proof: Let us consider the term (x^n - a^n) / (x - a).We can rewrite this expression in the form n.a^(n-1).(x-a/x-a). Now, since x tends to a, the limit of x-a/x-a is 1. Hence, the limit of (x^n - a^n) / (x - a) is equal to n.a^(n-1). Therefore, the limit of (x^n - a^n) / (x - a) when x tends to a is equal to n.a^(n-1), as desired.