Limit of a function f(x) at a particular point x, describes the behaviour of that function very close to that particular point. It does not give the value of the function at that particular point, it just gives the value function that seems to be taking on at that point.
The figure below presents a straight line and the limit of that function at point x = c, shown graphically.
Notice in the figure, as one approaches the point x = c from either left-hand side or right-hand side. The value of the function is changing. As one reaches point c, the value of the function seems to be converging to f(c).
Sometimes functions seem to be taking on some value in the limit and another value on the actual point. This usually happens in the functions which are not continuous and have jumped in their values.
For example, consider the function given in the figure below,
In this function, while approaching from the Left-Hand side, the function seems to be taking on the value, thus limit evaluates to L, but the actual value of the function at that point is f(c) = L’. Observe that, approaching from the right-hand side, the limit evaluates to f(c) = L’.
This proves that it is not necessary that the value of the function at a particular point is the same as the limit of the function at that particular point.
Let’s consider a function f(x), the function is defined on the interval that contains x = a. The limit of the function at x = a is denoted as, 
. This is written as “Limit of f(x) when x tends to a”.
Notice the words “tends to” or “approaches to” that are usually used to define the limits are not precise, and thus it is impossible to write that mathematically. So, a definition is needed which formalizes the notion for these phrases and words.
Consider the following function case,
The figure shows the values of a function f(x) at three different points which are close to each other. 
represents the change in the value of x, and the ∈ shows the change in the values the limit of the function at these points.
These parameters formalize the notion of the points being really close to each other and the meaning of the phrases like x approaching the point x = a. So, using the formal definition of the limits can be written down.
Epsilon-Delta Definition of Limits
The definition given below is called the “epsilon-delta” definition.
For the function f(x) defined on an interval that contains x =a. Then we say that,
If for every number epsilon(∈) which greater than zero, there is some positive number delta such that,
| f(x) – L | < ∈ where 0 < |x – a| <
Here,
- ϵ (epsilon) is a small positive number that represents how close we want the function’s value f(x) to be to L. Think of ϵ as the “tolerance” or “error margin” for how close we want f(x) to be to L.
- δ (delta) is another small positive number that depends on ϵ. It tells us how close x has to be to a in order for the value of f(x) to be within the ϵ-range of L.
For every small ϵ > 0, we can find a corresponding δ > 0 such that if x is within a distance of δ from a (but not equal to a), then the value of f(x) will be within a distance of ϵ from L.
How to Apply the Formal Definition
We can apply this formal definition of limits, to any case of limit using the following steps:
Step 1: State what you’re proving.
Step 2: Pick any small ϵ > 0.
Step 3: Find a corresponding δ > 0.
Let’s consider a function f(x) = 3x – 4.
Step 1: Here we can to proov 
.
This means we need to show that for every ϵ > 0, there exists a δ > 0 such that if 0 < ∣x − 2∣ < δ, then ∣(3x − 4) − 2∣ < ϵ|.
Step 2: Choose an arbitrary ϵ > 0.
Step 3: Simplify the expression for ∣f(x) − L∣ to find δ.
∣(3x−4) − 2∣ = ∣3x − 6∣ = 3∣x − 2∣
We need to make sure that 3∣x − 2∣ < ϵ/3. This happens when:
|x – 2| < ϵ/3
So, we choose δ = ϵ/3
Now we can conclude that if 0 < ∣x − 2∣ < δ, then:
∣(3x−4)−2∣ = 3∣x − 2∣ < ϵ
Therefore, by the ϵ−δ definition of limits, we have proven that:
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Solved Problems
Problem 1: For the given function f(x) = 5x -4, Prove using the epsilon delta definition of the limit, 
Solution:
For this case, L = 6 and a = 2. Consider any arbitrary number ∈ > 0.
|5x -4 – 6 | < ∈
⇒|5x -4 – 6| < ∈
The goal is to find a number , such that,
|5x -4 – 6| < ∈ where 0 < |x – 2| <
⇒ |5x -4 – 6| < ∈
⇒ 5 | x – 2| < ∈
⇒ |x – 2| < 
We can choose our = .
Now we can conclude that if 0 < ∣x − 2∣ < δ, then:
∣(5x−4)−6∣ = 5∣x − 2∣ < ϵ
Therefore, by the ϵ−δ definition of limits, we have proven that:
What do ϵ and δ represent?
- ϵ (epsilon) represents the desired tolerance for how close f(x) should be to the limit L.
- δ (delta) represents the range around ccc within which x must lie for f(x) to be within the tolerance ϵ.
When does the limit not exist?
The limit may not exist if:
- The function behaves erratically as x approaches c (e.g., oscillates infinitely).
- The function approaches different values from different directions (left-hand limit ≠ right-hand limit).
- The function grows without bound (e.g., tends to infinity).
How do you prove a limit using the epsilon-delta definition?
To prove that 
using the epsilon-delta definition, you:
- Choose an arbitrary ϵ > 0.
- Find a corresponding δ>0 such that for all x where 0 < ∣x − c∣ < δ, ∣f(x) − L∣ < ϵ.
Why is the epsilon-delta definition important?
The epsilon-delta definition provides a precise and rigorous way to describe limits. It helps ensure that our understanding of how functions behave as they approach a point is accurate and mathematically sound.
Can a function have a limit at a point where it is not defined?
Yes, a function can approach a certain value (limit) as x approaches a point, even if the function is not defined at that exact point.