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Imagine two construction crews building a bridge from opposite banks of a river. If both spans meet at the exact same elevation, cars can cross smoothly — but if one side ends three feet higher than the other, you get a cliff.
In calculus, functions can do the exact same thing when you approach a point from the left versus the right.

Approaching From One Side

A one-sided limit describes the value a function approaches as the input variable gets arbitrarily close to a target number from strictly one direction.
We write lim⁡x→c−f(x)\lim_{x \to c^-} f(x) for the left-hand limit (values of x<cx < c) and lim⁡x→c+f(x)\lim_{x \to c^+} f(x) for the right-hand limit (values of x>cx > c).
How do these two directional journeys determine whether a normal, two-sided limit actually exists?

The Two-Sided Limit Existence Theorem

In the 1820s, French mathematician Augustin-Louis Cauchy formalized directional approaches to eliminate confusion about what happens near boundary points.
The overall two-sided limit lim⁡x→cf(x)\lim_{x \to c} f(x) equals a real number LL if and only if both one-sided limits exist and equal that exact same number LL.
lim⁡x→cf(x)=L  ⟺  lim⁡x→c−f(x)=Landlim⁡x→c+f(x)=L\lim_{x \to c} f(x) = L \iff \lim_{x \to c^-} f(x) = L \quad \text{and} \quad \lim_{x \to c^+} f(x) = L
If the two paths arrive at different heights, the two-sided limit does not exist (abbreviated DNE).