# Limit

A limit is the behavior of a function as its inputs approach arbitrarily close to a given value.

## Intuitive Definition

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).

In other cases a limit can fail to exist, as approaching the limit from different sides produces conflicting values.

For cases in general, the expression below satisfies our needs. $\lim_{x \to a}f(x) = L$

The expression above states that when $x\,$ approaches arbitrarily close to $a\,$, the function $f(x)\,$ becomes arbitrarily close to the value $L\,$, which is called the limit.

## Rigorous Definition of a Limit

This definition is more appropriate for 2nd year calculus students and higher.

The original statement $\lim_{x \to a}f(x) = L$ now means that given any $\varepsilon > 0$, a $\delta > 0 \,$ exists such that if $0 < |x-a| < \delta \,$, then $|f(x)-L|< \varepsilon$.

We use the definition as follows to prove for the function $f(x) = 3x-1 \,$, $\lim_{x \to 2}f(x) = 5$

Given any $\delta\,$, we choose a $\varepsilon\,$ such that $\delta \leq \varepsilon/3$

If $|x-2|<\delta\,$, we can derive $|f(x)-5|<3\delta\,$.

$3|x-2|<3\delta\,$

$|3x-6|<3\delta\,$

$|f(x)-5|<3\delta\,$

$3\delta \leq 3*\varepsilon/3 = \varepsilon\,$

Thus $\lim_{x \to 2}f(x) = 5$

## Properties of a Limit

The limit of a sum of two functions is equal to the sum of the limits of the functions.

$\lim (f(x)+g(x)) = \lim f(x) + \lim g(x)$

The limit of a difference between two functions is equal to the difference between the limits of the functions.

$\lim (f(x)-g(x)) = \lim f(x) - \lim g(x)$

The limit of a product of two functions is equal to the product of the limits of the functions.

$\lim (f(x)*g(x)) = \lim f(x) * \lim g(x)$

The limit of a quotient of two functions is equal to the quotient of the limits of the functions (assuming a non-zero denominator).

$\lim (\frac{f(x)}{g(x)}) = \frac{\lim f(x)}{\lim g(x)}$