Limit and Continuity

## Limit and Continuity

In mathematics, a limit is the value that a function or sequence “approaches” as the input or index approaches some value.[1] Limits are essential to calculus (and mathematical analysis in general) and are used to define continuity, derivatives, and integrals.

The concept of a limit of a sequence is further generalized to the concept of a limit of a topological net, and is closely related to limit and direct limit in category theory.

In formulas, a limit is usually written as

$\lim_{n \to c}f(n) = L$

and is read as “the limit of f of n as n approaches c equals L“. Here “lim” indicates limit, and the fact that function f(n) approaches the limit L as n approaches c is represented by the right arrow (→), as in

$f(n) \to L \ .$
For example, if

$f(x) = \frac{x^2 - 1}{x - 1}$

then f(1) is not defined (see division by zero), yet as x moves arbitrarily close to 1, f(x) correspondingly approaches 2

In other words, $\lim_{x \to 1} \frac{x^2-1}{x-1} = 2$

This can also be calculated algebraically, as $\frac{x^2-1}{x-1} = \frac{(x+1)(x-1)}{x-1} = x+1$ for all real numbers x ≠ 1.

Now since x + 1 is continuous in x at 1, we can now plug in 1 for x, thus $\lim_{x \to 1} \frac{x^2-1}{x-1} = 1+1 = 2$

That means if you want to solve a Limit based question, you have to Factorise given function, then put the value.