Limit

Q1. What is the going away between a left field neighborhood and a decently neighborhood of a number? How does this creation become relevant in find a line of a intention? 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 infinitesim ally small. Similarly, recompense neighborhood of a number a represents numbers great than the number a and is denoted by a+ or or a+d, where d is infinitesimally small.This concept is very consequential in determining limit of a affair. A mold f(x) of x testament 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 die hard at a focalise of discontinuity does not exist. Why? Give an example.AnswerFor universe of limit of portion f(x) of x at x = a the necessary and competent condition is f(a-d) = f(a+d) = f(a) where d is infinitesimally small. At a acme of discontinuity, f(a-d) f(a+d).Therefore, limit of a function does not exist at a point of discontinuity. The following example volition make it clear. permit us sire example of integer function. This function is be in the following mannerf(x) = a where a is an integer less than or equal to x.Let us trail 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 sack up 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.AnswerDerivative of a function is another function, which remains analogous throughout the domain of the function at all the points. Slope of a function on the other hand is the cling to 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 magic spell derivative of a function remains the same while its slope could be changing from point to point.