Limit Definition of Derivative

The notion of a limit is an indispensable topic in Calculus of mathematics, yet it is also one of the most difficult. Calculus is a field of mathematics that deals with the computations required when dealing with constantly changing values. A function’s limit is when the function’s output approaches the specified input values. In general, there are two sorts of integrals: indefinite and definite. The upper and lower bounds of definite integrals are correctly specified. Indefinite integrals are stated without boundaries, and the function is integrated using an arbitrary constant that may be chosen at any time. 

Definition of the limit of a function

Let’s learn about the definition of the limit of a function. 

Assume L is a real number and that f(x) has all possible values inside an open interval, including as an exception. When f(x) approaches the real number L, we refer to it as the limit of f(x) when the number an approaches x is not equal to a. In other words, as x moves closer and closer to a, f(x) remains near to L.  We can represent this in the following manner: 

limx→a f(x)=L

So, this is the definition of the limit of a function. 

Properties of Limits

After studying the definition of the limit of a function. Let’s learn some properties of limits are:

• It is possible to indicate the limit of a function by the notation f(x) = L when x approaches the limit of the function, with the result that

  limx→af(x) = L

• A function’s limit is equal to the sum of its limits, such that: limx→a [f(x) + g(x)] =  limx→a f(x) + limx→a g(x)

• The limit of any constant function is a constant, such that limx→a C = C is true for every constant function.

• The Quotient Rule states that limx→a[f(x)/g(x)] =  limx→af(x)/limx→ag(x); if this condition is valid limx→ag(x) ≠ 0

• Because the constant limit and the function limit are equivalent, we may write this as: limx→a m f(x) = m limx→a f(x)

Limit definition of derivative

Let’s learn about the limit definition of a derivative. 

In a nutshell, the derivative of a function tells us how fast a function is changing. Consider the following scenario: you’re travelling in the car on a  highway, and the function f(t) calculates the distance (in miles) from your starting position as a function of the time t (in hours) since you began your journey.

A glance at the speedometer on the dashboard of your vehicle, say two hours into the journey, will reveal the speed at which your vehicle is driving (in miles per hour) at that time. The derivative of a function f provides us with the following information: f'(2) at t=2.

When t = 2, the instantaneous rate of change of f is denoted by f’(2). If we know the derivative, we can calculate or estimate many useful features of the car’s velocity at any given point in time, t.

Definite integral 

In situations when the boundaries are specified, and it is necessary to obtain a unique result, definite integrals are utilised. A definite integral is described as having both upper and lower bounds and does not have any intermediate values.

The definite integral is represented as:     ∫ab f(x) dx = F(b) – F(a)

Definite integral as a limit of sum

Let’s learn Definite integral as a limit of sum

Consider a curve that lies above the x-axis. There are no negative values for the function in this graph, which is defined as an infinite continuous function on the closed interval [a, b]. In the case of continuous function f, the area bound between the curve, the points ‘x = a and b,’ and the x-axis is equal to the integral of the definite integral of the function f.

The area of the region encompassed by the curve y = f(x) is defined as the integral of f(x). The region ABCD depicts this area. This whole area between [a and b] is subdivided into n equal subintervals, each of which is defined by [x0, x1], [x1, x2],…… [xr-1, xr], [xn-1, xn]

Assuming that h 0, the width of each subinterval, and x0 = a , x1 = a + h, x2 = a + 2h,…,xr = a + 2h,…,xn  = a + nh.

as well as n = (b – a)/h

We may infer that the limiting values of sn and Sn are equivalent since the rectangular strips are quite thin. The mutual limiting value provides us the area  under the curve.

lim Sn n→ ∞ = lim sn n→ ∞ = Area of the region ABCD = f(x)dx

Therefore, 

∫ab f(x) dx = limn → ∞ h [f(a) + f(a + h) + …. + f(a + {n – 1}h)]
Or, ∫ab f(x) dx = (b – a) limn → ∞ (1/n) [f(a) + f(a + h) + …. + f(a + {n – 1}h)] 

In this case, h = (b – a)/n →0 as n→∞

Hence, this is the formula for finding the definite integral as limit of sum.

Conclusion 

Definition of limit of a function is considered to be one of the most important fields of mathematics. It is a method of solving issues systematically that is generally concerned with determining the attributes or values of functions via the use of integrals and derivatives. Differentiation and integration are the fundamental concepts of calculus. The two notions might be thought of as being diametrically opposed to one another in their definitions. The differential is the inverse of integral, and the inverse of integral is differentiable. Integrals are classified as either definite or indefinite in terms of the consequences they produce.