Some rules:
If $\displaystyle \lim_{x \to a} f(x) = L$ and $\displaystyle \lim_{x \to a} g(x) = M $
then $\displaystyle \lim_{x \to a} (f+g)(x)=L+M$ and $\displaystyle \lim_{x \to a} (fg)(x)=LM $
also $\displaystyle \lim_{x \to a} \ (kf)(x) = kL $
Prove that $\displaystyle \lim_{x \to a}x = a $
Now use the product rule and induction to show that $\displaystyle \lim_{x \to a}x^{n} = a^{n} $
Finally use the mutiple rule and the sum rule