# Limit

Q1. What is the difference between a left neighborhood and a right neighborhood of a number? How does this concept become relevant in determining a limit of a function?

Left neighborhood of a number ‘a’ represents numbers lesser than the number ‘a’ and is denoted by ‘a-’ or ‘a-d’, where d is infinitesimally small. Similarly, right neighborhood of a number ‘a’ represents numbers greater than the number ‘a’ and is denoted by ‘a+’ or or ‘a+d’, where d is infinitesimally small.

This concept is very important in determining limit of a function. A function f(x) of ‘x’ will have a limit at x = a; if and only if f(a-d) = f(a+d) = f(a); where d is infinitesimally small.
Q2. A limit of a function at a point of discontinuity does not exist. Why? Give an example.

For existence of limit of function f(x) of ‘x’; at x = a; the necessary and sufficient condition is f(a-d) = f(a+d) = f(a); where d is infinitesimally small. At a point of discontinuity, f(a-d) ≠ f(a+d).

Therefore, limit of a function does not exist at a point of discontinuity. The following example will make it clear.

Let us take example of integer function. This function is defined in the following manner:

f(x) = a;           where ‘a’ is an integer less than or equal to x.

Let us check if limit exists for this function at x = ‘a’, where ‘a’ is an integer.

Now left hand side limit = f(a-d) = a-1

And right hand side limit = f(a+d) = a

Thus, f(a-d) ≠ f(a+d); and hence limit does not exists for this function. If this function is plotted, there is discontinuity at all integer points.

Thus it can be seen that limit of a function does not exist at a point of discontinuity.
3. What is the difference between a derivative of a function and its slope? Give a detailed explanation.