The Power of Limits in Calculus

Calculus is built on the concept of limits. Calculus' other fundamental concepts, such as differentiation and integration, can only be comprehended by knowing how to perform limits. A limit is a value that f(x)f(x) approaches as xx approaches some value when given a function ff.

Take, for instance, the function f(x) = 2xf(x)=2x. Let's say we want to determine the limit of the function ff at x = 2x=2. The yy-value that ff approaches when xx gets infinitely close to 2 is what we want to find. It is evident from the graph that yy is getting closer to 4 as xx gets closer and closer to 2. As xx gets closer to 2, we say that the limit is 4 in this case.

It is essential to make it clear that the concepts of a function's limit at xx and its value at xx are distinct. In the preceding example, the limit of ff as xx approaches 2 equals the value of ff at 2. This isn't always the case, though. The difference is that even when ff is not defined at cc, the limit as xx approaches some cc can frequently be evaluated.

Consider, for instance, the graph below for the function f(x) = fracx2-1x-1f(x)=x1 x 2 1. At x = 1x=1, there is a discontinuity that can be removed, which is called a hole because we can't have a denominator that is zero.
In calculus, why do we use limits? Because they enable us to examine the behavior of a function around a particular x-value even when the function is not defined at that x-value, limits are useful in calculus.

Calculus also relies on limits as its foundation. The foundations of calculus were initially based on infinitesimals, which were eventually deemed to be too vague to be useful. In calculus, limits were created to replace these vague arguments with arguments that were more mathematically precise. Now, the most important concepts in calculus are defined by limits.
The derivative of the function ff at xx is said to be LL if the limit LL is present, and the expression "frac" can be written as "dx d f(x)=L." This limit is the instantaneous rate of change of f at xx.

Integrals can also be defined using limits.

In this notation, "limlim" denotes the operation of taking a limit, so if we want to evaluate the limit of a real-valued function ff as xx approaches some real number cc, we can use f(x)=L. The value that xx approaches, at which we want to evaluate the limit, is represented by "x to cxc" beneath. Finally, as f(x)f(x) gets closer to cc, LL represents the limit solution.