A limit at x=c exists when:

For a limit to exist at a specific point $x = c$, the function must approach the same finite value from both directions. 1. **The Left-Hand Limit**: This is the value the function approaches as $x$ gets closer to $c$ from values smaller than $c$. It is denoted as $\lim_{x \to c^-} f(x)$. 2. **The Right-Hand Limit**: This is the value the function approaches as $x$ gets closer to $c$ from values larger than $c$. It is denoted as $\lim_{x \to c^+} f(x)$. **The Formal Condition:** A limit $\lim_{x \to c} f(x)$ exists and equals $L$ if and only if: $$\lim_{x \to c^-} f(x) = \lim_{x \to c^+} f(x) = L$$ It is important to note that $f(c)$ does not need to be defined for the limit to exist. A function can have a 'hole' at $x=c$ (a removable discontinuity), and the limit will still exist as long as the paths from the left and right meet at that hole.