Limit of a Function at a Point

Limit of a Function at a Point

• A function $f(x)$ has a limit as $x$ approaches $c$ if and only if both the left-hand limit and the right-hand limit exist and are equal.

• This can be expressed notationally as follows:

  $\lim<em>{x \to c} f(x) \text{ exists } \iff \lim</em>{x \to c^-} f(x) = \lim_{x \to c^+} f(x)$

  • Where:

    • $\lim_{x \to c^-} f(x)$ represents the left-hand limit (as $x$ approaches $c$ from the left).
    • $\lim_{x \to c^+} f(x)$ represents the right-hand limit (as $x$ approaches $c$ from the right).

• In essence, for the limit of a function to exist at a point, the function must approach the same value from both the left and the right sides of that point.