What is a Limit?
The limit is the main concept for calculus before the learning of integral and derivative calculus. Because the limit is the most useful concept in both integral and derivative calculus. The limit is not a small concept to explain, but in simple words limit is the occurrence of continuous change.
So here we elaborate that what is a limit of a function with examples. So by these examples, you can not only learn the concept of the limit but also able to know precise definition of limits, its importance and its usage.
Example
Let’s discuss the area of the rectangle, most probably all of us know that:
$$\text{Area of a rectangle = Length * Width}$$
Also, we know about the area of a circle,
$$ \text{Area of circle} = πr^2 $$
But most of us can’t describe the proof of that round figure area. The circle is not a rectangular area figure and it takes a long time for scientists to determine the area of curve figures.
Here we will prove it in four simple steps.
In the very first step, we will divide a circle into four equal pieces and rearrange them in such a way.
You can see a second disordered shape that is formed by the same circle, and both these shapes have the same area. The curve area of each chop is the circumference of the circle i.e 2πr.
We can see in the above figure the two curve arcs are equals to half of the circumference of a circle which is πr.
In step 2, Now we repeat the same experiment in 8 equal chops in this way it looks some meaningful than the old one.
Now you can see a second less disordered shape that is formed by the same circle. You can see the same result but half of the circumference contains now 4 arcs equal to πr.
But now we divide that circle into infinity equal chops, now it looks more meaningful than the last one.
In this way, the shape looks more meaningful and close to a rectangle shape. Now the rectangle is discovered and the area of a rectangle can easily be calculated.
Now infinity of curve arcs is equaled to the length of the rectangle which is πr and the height or width of that rectangle is also R. So, the area of the circle is determined by simple steps following.
We know the term infinity is not actually happened or possible but it may be supposed for resolving big problems. The continuous change in a circle to disordered shapes will form it more likely to the area of the rectangle. This is the same concept of limit which resolves the term infinity.
General concept:
So, we see the basics of limits and approaching. We analyze from where the concept of limit is discovered and how scientists find this concept of limit from rectangle to circle by approaching closer and closer up to infinity.
Now, we see the precise definition of limit in terms of mathematics.
Precise Definition of Limits:
The definition of limit of a function helps us to understand that what is limit in maths. So let's see the the precise definition of limits.
Let f(x) be a function that is defined on an open interval about “c”, which may be perfectly c itself. In this condition,
$$ \text{The Limit f(x) as “x” approaches to “c” is the number L}$$
It means if x approaches to c(not exact c) then f(x) approaches to L. In this way, limit definition formula may written as:
$$ \lim\limits_{x \to c} f(x) = L$$
If the number Epsilon > 0 , then the corresponding number exists are also Delta > 0 such that for all values of x.
Epsilon and Delta are reated to each other in the definition of limit of a function. The Epsilon Delta definition of limit is expressed in a graph as follow:
Epsilon and Delta:
In the above graph, we can see if x is approaching c as
$$c-\delta < x < c+\delta, $$
Then delta is a very small distance which is very very close to c or approaching toward c.
Similarly, as x lies between
$$c-\delta \;\; and \;\; c+\delta $$
It means the function f(c) also lies between
$$\text f(c - \delta) \;\; and \;\; f(c + \delta)$$
In this way, Epsilon Delta definition of limit can be written as:
$$ f(c - \delta) = L + \epsilon \;\; and \;\; f(c- \delta)=L+\epsilon$$
We can see as the Delta is a very small distance from a certain value of x=c. Similarly, Epsilon is also a very smaller distance from function at that value f(c).
In general, Epsilon and Delta are approaching c and f(c) which is called the definition of limit of a function.
So, We have learned the concept that what is limit in maths and definition of limit of a function which helps us a lot in our future topics of limits which makes calculus easy to understand.