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.
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.
Derivative of a function is another function, which remains same throughout the domain of the function at all the points. Slope of a function on the other hand is the value of the derivative. This value may change from point to point depending on the nature of the function.
Let us take an example. Derivative of Sin(x) is Cos(x) for all values of ‘x’. If one looks at the slope of Sin(x), its value keeps changing in [-1, +1] range from point to point. Slope of Sin(x) is -1 for x = odd integral multiples of p; +1 for x = even multiples of p and 0 for x = odd multiples of p/2. Thus, it can be seen that while derivative of a function remains the same while its slope could be changing from point to point.