Limits

Limits are the locations where a function goes toward the final result for the given input values in mathematics. Limits are used in calculus and mathematical analysis to define things like continuity, integrals, and derivatives among other things. It is employed in the analysis process and always refers to the function’s behaviour at a specific time. 

The concept of a sequence limit is further developed in the concept of a topological net limit, which is related to the theory category’s limit and direct limit. Integrals are generally divided into two categories: definite and indefinite integrals. The upper and lower limits of definite integrals are properly defined. When integrating the function, indefinite integrals are expressed without limits and have an arbitrary constant. 

Properties of Limits

There are numerous principles you may use to discover the limit of a function to make it easier to find the limit.

1. Sum Rule

The sum rule asserts that the limit of the sum of two functions, f(x) + g(x), as x approaches a is the same as the limit of f(x) plus the limit of g(x).

2. Constant Rule

Constant functions are involved in the constant rule. Any function that has only one constant, c, is referred to as a constant function.

3. Product Rule

As x approaches a value an of the product of two functions, f(x) * g(x), the product rule asserts that the limit as x approaches value an of f(x) multiplied by the limit as x approaches value an of g(x) is the same as the limit as x approaches value an of f(x).

Algebra of Limits

To understand the limits, let’s consider an example like p and q to be two functions such that their limits limx→a p(x) and limx→a q(x) exists.

• The sum of the limits of two functions is the sum of their limits.

limx→a [p(x) + q(x)] = limx→a p(x) + limx→a q(x).

• The difference of the limits of two functions is the difference of their limits.

limx→a [p(x) − q(x)] = limx→a p(x) − limx→a q(x).

• The product of the limits of two functions is the limit of the product of the limits of the functions.

limx→a [p(x) × q(x)] = [limx→a p(x)] × [limx→a q(x)].

• The quotient of the limits of two functions is the limit of the quotient of the limits of the functions.

limx→a [p(x) ÷ q(x)] = [limx→a p(x)] ÷ [limx→a q(x)].

• The limit of a function p(x) multiplied by a constant, q(x) = is times the limit of p (x).

limx→a [α.p(x))] = α. limx→a p(x).

Limit of Polynomial Function

Consider a polynomial function, f(x) = a0 + a1x + a2x2 + … + anxn. Here, a0, a1, …, an are all constants. At any point x is equal to a, the limit of this polynomial function is

limx→a f(x) = limx→a [a0 +a1x + a2x2 + … + anxn]

= limx→a a0 +a1 limx→a x + a2 limx→a x2 + … + an limx→a xn

or, limx→a = a0 +a1a + a2a2 + … + anan = f(a).

Limit of Rational Function

Let’s consider the rational function’s limit of the type p(x) / q(x), where q(x) ≠ 0 and p(x) and q(x) are polynomial functions

limx→a [p(x) / q(x)] = [limx→a p(x)] / [limx→a q(x)] = p(a)/q(a).

The first step in determining a rational function’s limit is to see if it can be reduced to the form 0/0 at some point. If this is the case, several adjustments must be made to calculate the value of the limit. 

The factor that causes the limit to be of the type 0/0 is cancelled.

Conclusion

The theory of calculus is built on the concept of limits. Limits of functions are used to construct derivatives of functions, verify for function continuity, and so on. The value of a function’s limit at a certain moment offers us an intuitive impression of the function’s approaching value. It’s worth noting that when we calculate the limits, we’re not looking for the precise value of the function at that specific point.