Limit of a function

In Mathematics, limit is defined as a value that a function approaches for the given point. It is always concerns about the behavior of a function at a particular point.

Meaning of \( x\to a \)

Consider a function
\(f(x)= \frac{x^2-1}{x-1}\)
We know that function is NOT defined at x=1. However, what happens to f(x) near the value x=1?

If we substitute small values for x, then we find that the value of f(x) is approximately 2 near at x=1

	x<1						x>1
x	0.5	0.6	0.7	0.8	0.9	1	1.1	1.2	1.3	1.4	1.5
f(x)	1.5	1.6	1.7	1.8	1.9	2	2.1	2.2	2.3	2.4	2.5

The closer that x gets to 1, the closer the value of the function f(x) to 2.
In such cases, we call it
f(x)=2 as x tends to 1

Intuitive Definition of Limit

Let 𝑓(𝑥) be a function defined at all values in an open interval containing a, with the possible exception of "a" itself, and let L be a real number.
Then we say that \( \displaystyle \lim_{x \to a} f(x) =L\) if
for a given number \(\epsilon > 0\), there exists a number \(\delta > 0\)such that
|f(x)-L|<\(\epsilon\) whenever |x-a|<\(\delta\)

In this definition

1. Given a number L, we choose ε-neighbourhood of L, ε is positive AND can be small enough as we like such that |f(x) - L|< ε
2. Now, we will try to find, δ-neighbourhood of a, δ is positive AND can be small enough as satisfied such that |x - a|< δ
3. If a small change in ε implies a small change δ, then the limit exists at a.
4. If a small change in ε implies a LARGE change δ, then the limit does NOT exist at a.

Show δ-neighbourhood of "a"
Show ε-neighbourhood of "L"

How to use the applet

1. Click on the "Show δ-neighbourhood of 'a'" check box.
2. Given a number a, adjust δ-neighbourhood of a (drag the point a or the slider δ) , so that |x - a|< δ , where x is any point inside the δ-neighbourhood
3. Now, Click on the "Show ε-neighbourhood of 'L'" check box.
4. Try to find ε-neighbourhood of L (largest distance from L) , such that |f(x) - L|< ε , where f(x) is any point inside the ε-neighbourhood and the result is valid for all |x - a|< δ
   
   सबै |x - a|< δ को लगी |f(x) - L|< ε  हुने गरि ε-zone बानउन सकिन्छ भने limit exist हुन्छ ।
   
   Once a ε is found, any higher ε is always accepted.
   Once a δ is satisfied, any smaller δ is always accepted.

More Explanation

The intuitive definition says that

1. determine a number δ>0
2. take any x in the region, i.e. between a+δ and a−δ, then this x will be closer to a, that is |x-a|<δ
3. identify the point on the graph that our choice of x gives, then this point on the graph will lie in the intersection of the ε region. This means that this function value f(x) will be closer to L , that is |f(x)-L|<ε
   Means
   if we take any value of x in the δ region then the graph for those values of x will lie in the ε region.
4. Once a δ is found, any smaller delta is acceptable, so there are an infinite number of possible δ's that we can choose.
5. the function has limit at given x

Empirical Definition of Limit

Let 𝑓(𝑥) be a function defined at all values in an open interval containing a, with the possible exception of "a" itself, and let L be a real number.
Then we say that \( \displaystyle \lim_{x \to a} f(x) =L\) if
all of the following three conditions hold

1. \( \displaystyle \lim_{x \to a^{-}} f(x) \) exists=LHS
2. \( \displaystyle \lim_{x \to a^{+}} f(x) \) exists=RHS
3. LHS=RHS
1. In Figure (1). We see that the graph of f(x) has a hole at a. In fact, f(a) is undefined.[Limit exists at x=2]
2. In Figure (2), f(a) is defined, but the function has a jump at a.[Limit does NOT exist at x=2]
3. In Figure (3), f(a) is defined, but the function has a gap at a.[Limit exists at x=2]

\( f(x)=\frac{x^2-1}{x-1}\)	\( \small {f(x)= \begin{cases} x+1 & \text{for } x ≤ 2 \\ x+2 & \text{for } x > 2 \end{cases}} \)	\( \small { f(x)= \begin{cases} x+1 & \text{for } x \ne 2 \\ 4 & \text{for } x = 2 \end{cases}} \)