If two one sided limits exist, the two sided limit exists.

Question

prove that if the limit as x approaches a from the positive side of the x axis, f(x) = L, and if the limit as x approaches a from the negative side of the x axis, f(x) = L, then the limit as x approaches a of f(x) = L as well.

Answer 1

You titled this "If two one sided limits exist, the two sided limit exists." It should be "If two one sided limits exist, and are equal, the two sided limit exists."

In order to show that $\lim_{x\to a} f(x)= L$, you want to show that, for any $\epsilon> 0$ there exist $\delta> 0$ such that if $|x- a|< \delta$ then $|f(x)- L|< \epsilon$.

Since $\lim_{x\to a^-} f(x)= L$ we know that, for any $\epsilon> 0$ there exist $\delta^-> 0$ such that if $x< a$ and $a- x< \delta^-$ then $|f(x)- L|< \epsilon$.

Since $\lim_{x\to a^+} f(x)= L$ we know that, for any $\epsilon> 0$ there exist $\delta^+> 0$ such that if $x> a$ and $a- x< \delta^+$ then $|f(x)- L|< \epsilon$.

So, given $\epsilon> 0$, take $\delta$ to be the smaller of $\delta^-$ and $\delta^+$ so that if $|x- a|< \delta$ then both $|x- a|< \delta^-$ and $|x- a|< \delta^+$ are true. Whether $x> a$ or $x< a$, $|f(x)- L|< \epsilon$.