Limit

Q1. What is the difference among a left neighborhood and a remunerate neighborhood of a number? How does this concept become relevant in determining a barrier of a function? AnswerLeft neighborhood of a number a represents numbers lesser than the number a and is de noned 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 signifi chamberpott 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 transfer of discontinuity does not exist. Why? Give an example.AnswerFor origination of limit of function f(x) of x at x = a the obligatory and sufficient condition is f(a-d) = f(a+d) = f(a) where d is infinitesimally small. At a prefigure of discontinuity, f(a-d) f(a+d).Therefore, limi t of a function does not exist at a point of discontinuity. The following example will make it clear.let us direct example of integer function. This function is defined in the following mannerf(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-1And right hand side limit = f(a+d) = aThus, 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.Answer derivative of a function is another function, which remains same throughout the dry land of the function at all the points. Slope of a function on the other hand is the protect of the derivative. This value may change from point to point depen ding on the nature of the function.Let us take an example. first 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 = laughable 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.