What are Limits?
The limits in math is not a small concept to explain, but in general, the limit of a function is the occurrence of continuous change and approaching toward closer to something.
But the question is what are limits in math?
Let's try to understand the definition of a limit by using limit of a function examples.
Let’s say we have a function
$$ f(x)= \frac{x^2-4}{x-2} $$If we have to find limit of a function at f(2), we simply have to substitute x by 2
Now,
What is that 0/0 form?
Now we do not know what are limits at infinity or at 0/0. It could be 0? it could be infinity? it could be undefined or something else we do not know. So the function is not defined at 2.
When x = 2 we do not have a specific value. However, as x approaches closer to 2 it does converge to a specific value.
Let's plug in a number that is not exactly 2 but something that's close to 2.
Let's make a table for limit of a function
| x | $$\frac{x^2-4}{x-2}$$ |
|---|---|
| 1.9 | 3.9 |
| 1.99 | 3.99 |
| 1.999 | 3.999 |
| 1.9999 | 3.9999 |
So, notice that as x approaches 2 the expression x2-4/x-2 approaches the number 4. It gets closer and closer to 4. So therefore the limit is 4.
So, it can never be wrong to say that by using the special concept of “limit” we may answer 0/0 in some appropriate way.
As we clap our hands both hands need to meet at some distinct point. For a minute, If we consider our limit of a function the same distinct point we need to evaluate the work of both sides.
Let’s do it the same from another side
Let’s make a table for limit of a function from other side:
| x | $$\frac{x^2-4}{x-2}$$ |
|---|---|
| 2.01 | 4.01 |
| 2.001 | 4.001 |
| 2.0001 | 4.0001 |
| 2.00001 | 4.00001 |
It also approaches toward the same limit of 4 from another side. So, its ok to say the limit is 4 for a given function.
But now the question arises what happens if the results from sides approaching are not the same?
When both are different?
In the case of similar approaches from both sides, there may be a discontinuous curve graph obtained with an unfilled hole at point 4. Because both will approach closer and closer to it.
But if both are not matching there will be a discontinuous curve graph with a gap at a certain point. In such a case limit does nott exist at such a point.
In a given graph
- The limit is “1.5” from the left side.
- The limit is “4” from the right side.
But in fact, limit of a functon does not exists at that point.