Uniqueness of Limits and Calculus of Limits

Theorem: CCD.UL: If $\lim_{x \to a}f(x)= L$ and $\lim_{x \to a}f(x)= M$ then $L=M$.
Proof of CCD.UL

Theorem: CCD.CL: Suppose $\lim_{x \to a}f(x)= L$ , $\lim_{x \to a}g(x)= M$,  and $\alpha$ is a real number,then
(i) $\lim_{x \to a} \alpha f(x)= \alpha L$;
(ii) $\lim_{x \to a} f(x) \pm g(x) = L \pm M$;
(iii) $\lim_{x \to a} f(x) \cdot g(x) = L \cdot M$;
(iv) If $M \ne 0$,  then $\lim_{x \to a} \frac {f(x)}{g(x)} = \frac LM$;
(v) If $\lim_{y \to L} g(y)= N$ and $g(L)=N$, then $\lim_{x \to a}g(f(x))= N$.

Discussion and Some Proofs of CCD.CL