What is the concept of a limit in calculus?

In calculus, limits are extremely important because calculus itself is based upon limits. Basically, a limit describes the behavior of a dependant variable when its independent variable takes extreme values.

For example, lets consider this function : y =1/x. As you know, this typical function is not defined at x = 0 because the division by zero is not admitted in real numbers. Therefore we cannot compute the value of y when x = 0. However, we can observe how the function behaves near x = 0 : this is the concept of limit. Lets see how this function behaves when x approaches zero from the right:

limx->0+ 1/x = infinity

You can verify this limit by substituting x with values that approach zero: 1/0.1 = 10; 1/0.000009 = 11,111.11; ... When x takes extremely small values, y takes extremely large values. If we repeat this process forever, we will say that, at the limit, y will have an infinite value. That's what the limit is.

Two of the most important uses of limits in calculus are derivatives/differentials and integrals. For example, the derivative of a function is the limit of the function's ratio of variation of dependant and independent variables as the variation of the independent variable approaches zero:

derivative of f(x) with respect to x = f'(x) = d[f(x)]/dx ...

Also, a definite integral is defined as the limit of the sum of infinitely small elements as the number of elements approaches infinity.

To calculate limits, you must have a good knowledge of the "algebra of infinity" (infinity + infinity = infinity, sqrt(infinity) = infinity, ...).Of course, this is a very basic description of the limit concept; there are many, many cases where the limit is tricky to calculate.

The concept of Functions limits and Continuity leads to define and describe continuity and derivative of the function.

The continuity of a function has practical as well as theoretical importance. We plot graphs by taking the values generated in the laboratory or collected in the field. We connect the plotted points with a smooth and unbroken curve (continuous curve). This continuous curve helps as to estimate the values at the places where we haven't measured. It was developed by Isaac newton and Leibnitz.

Here in this chapter, we will study some standard functions, their graphs, concept of limits and discuss about the continuity of the functions. Throughout this chapter, we denote R as the set of real numbers.

Types of Limit:

Left Hand Limit: Let f(x) tend to a limit l1 as x tends to a through values less than 'a', then l1 is called the left hand limit.

Right Hand Limit: Let f(x) tend to a limit l2 as x tends to 'a' through values greater than 'a', then l2 is called the right hand limit.

We say that limit of f(x) exists at x = a, if l1 and l2 are both finite and equal.