formal definition of limit and approach from both sides The formal definition of limits defines when a limit should exist (as seen here Wikibooks entry
In addition, I've seen from several sources that a limit exists only when the function approaches the same value from both the left and right sides. 
How do I tie these two together? In other words, how does the formal definition of limits ensure that a limit exits only when the function approaches the same value from both sides? 
 A: As mentioned in my comment,Right hand limit checks for $0<x−c<δ$. Left hand limit checks for $−δ<x−c<0$. A two sided-limit checks for $−δ<x−c<δ$. To see why this matters, observe the limit $\lim_{x \to 0}{1 \over x}$.
Left-hand limit:
By definition, I can only choose $x$ in $(-\delta,0)$. Thus, every value I choose for $x$ will result in $1\over x$ being a point in the curve that approaches $-\infty$ from the left side.
Right-hand limit: In a similar way, every value I choose for $x$ is in $(0, \delta)$, so $1\over x$ will always be a point in the curve that approaches $+\infty$ from the right side.
Two-sided-limit: Here's where the problem lies. I can pick an $x$ from either the right-hand curve or the left-hand curve because I'm choosing $x$ from $(-\delta, \delta)$. There is no distinct "one limit" that $x$ is approaching to given the interval of choice. The confusion that ensues results in the conclusion that no limit exists given the formal definition of a "two-sided" limit. 
On the other hand, assume that both a right-hand limit and left-hand limit exist for a function $f$ at $c$. Then we have:
For every $\varepsilon > 0$, there exist $\delta_1 > 0$ and $\delta_2 > 0$ such that  $-\delta_1 < x - c < 0$ and $0 < x - c < \delta_2$ imply that $|f(x) - L| < \varepsilon$. Thus, there exists $\delta = min(\delta_1, \delta_2) > 0$ such that $0 < |x - c| < \delta$ imply that $|f(x) - L| < \varepsilon$, so by definition, $f$ has a two-sided limit.
From the above paragraph we can conclude that all functions whose right-hand and left-hand limits match must have a two-sided limit that exists (and is equal with either side's limit).
