The limit of a function f(x) exists at a point x₀ if and only if the following two conditions are met:

1. The limit from the left exists:

   lim _x→x₀^- f(x) exists and is a finite number. This means as x approaches x₀ from values less than x₀, the function approaches a specific value.

2. The limit from the right exists:

   lim _x→x₀⁺ f(x) exists and is a finite number. This means as x approaches x₀ from values greater than x₀, the function approaches a specific value.

3. The left-hand limit and the right-hand limit are equal:

   lim _x→x₀^- f(x) = lim _x→x₀⁺ f(x)

   If both the left-hand and right-hand limits exist and are equal, then we can say the limit of f(x) as x approaches x₀ exists and is equal to that common value. Mathematically:

   lim _x→x₀ f(x) = L

In simpler terms, for a limit to exist at a point, the function must approach the same value as it gets closer to that point from both sides.