WHAT IS LIMIT? In mathematics, the concept of a "limit" is used to describe the value that a function or sequence "approaches" as the input or index approaches some value. [1] The concept of limit allows mathematicians to define a new point from a Cauchy sequence of previously defined points within a complete metric space. [2] 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, limit is usually abbreviated as lim as in lim(an) = a or represented by the right arrow () as in an a.Limit of a functionLIMIT OF A FUNCTION Limits are a mathematical tool which is used to define the 'limiting value' of a function i.e. the value a function seems to approach when it's argument(s) approach a particular value. Although, the argument of the function can be taken to approach any value, limits are helpful in cases where the argument approaches a value where the function is not defined or becomes exceedingly large.While defining a limit, we say that the argument 'tends to' a value. For example,limxcf(x)=mis said: "As x tends to c, the function f(x) tends to m". This statement however makes no assertion of what the value of f(c) would be. Rather, it means that as x becomes exceedingly close to 'c', f(x) becomes exceedingly close to m.If, however, the function is defined and continuous at 'c', then:limxcf(x)=f(c)EXAMPLE x.x - 5x + 6 (x - 2)(x - 3) lim --------------- = lim --------------- = 1 3 x - 3 3 x - 3 lim sqrt(x) is not defined -2 2.x2 + x 2 lim ------------ = --- +infty 3.x2 + 4 3 LEFT AND RIGHT HAND In some cases, you let x approach the number a from the left or the right, rather than "both sides at once" as usual. means: Compute the limit of as x approaches a from the right. means: Compute the limit of as x approaches a from the left.The left- and right-hand limits are the same if and only if the ordinary limit exists. In this case, the left-hand, right-hand, and ordinary limit are equal.Example.Look at the first limit more closely. x approaches 0 from the right. Numbers close to, but to the right of, 0 are small positive numbers: 0.01, for example. Small positive numbers make positive: , for example. If is positive, then , so(Notice that you don't let x equal 0, so , and the cancellation is legal.)Therefore,Here's the picture:HOW TO USE LIMIT IN CONTINUITY OF A FUNCTION?The following problems involve the CONTINUITY OF A FUNCTION OF ONE VARIABLE. Function y = f(x) is continuous at point x=a if the following three conditions are satisfied :i.) f(a) is defined ,ii.) exists (i.e., is finite) ,andiii.) Function f is said to be continuous on an interval I if f is continuous at each point x in I.EXAMPLEExamine the continuity of the functionIf , we call the limit of this function as x approaches 1 from the left asSimilarly, we call the limit as x approaches 1 from the right equal to 2. The function obviously has a value of 2 when x = 1.We say that