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.

SimpleParabola.png

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).