# Limit

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A limit is the behavior of

## Intuitive Definition

We know that a function assigns values to outputs with respect to inputs. As given inputs get closer and closer to a specific value, the function's output get closer and closer to a certain separate value. We define this value that the function's output approaches as the limit.

We can examine the limit of a simple continuous function, $f(x)=x^2 \,$.

The limit at x=0, would simply be f(x)=0, since as the function is continuous and differentiable at every point, $f(0)=0$.

Indeed for this function, we define the limit as $\lim_{x \to a}f(x) = f(a) \,$. But this is a special case of limit, in the majority of limits cannot be solved in this manner.

For example, given

$f(x)=\left\{\begin{matrix} {x}^2, & \mbox{if }x\ne 0 \\ \\ 1, & \mbox{if }x=0. \end{matrix}\right.$

The limit of $f(x)\,$ as x approaches 0 is 0 (just as in $f(x)\,$), but $\lim_{x\to 0}f(x)\neq f(0)$; $f(x)$ is not continuous at $x = 0$ (as shown on the right).