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

- 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
- 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
- If a small change in ε implies a small change δ, then the limit exists at a.
- If a small change in ε implies a LARGE change δ, then the limit does NOT exist at a.

Show ε-neighbourhood of "L"

## How to use the applet

- Click on the "Show δ-neighbourhood of 'a'" check box.
- 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
- Now, Click on the "Show ε-neighbourhood of 'L'" check box.
- 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- determine a number δ>0
- 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
- 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. - Once a δ is found, any smaller delta is acceptable, so there are an infinite number of possible δ's that we can choose.
- the function has limit at given x
- \( \displaystyle \lim_{x \to a^{-}} f(x) \) exists=LHS
- \( \displaystyle \lim_{x \to a^{+}} f(x) \) exists=RHS
- LHS=RHS
- 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]
- In Figure (2), f(a) is defined, but the function has a jump at a.[Limit does NOT exist at x=2]
- In Figure (3), f(a) is defined, but the function has a gap at a.[Limit exists at x=2]

#### 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

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

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