# Limits, Continuity and Differentiability: What Are Limits: Existence of Limit (For CBSE, ICSE, IAS, NET, NRA 2022)

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# Title: Limits, Continuity and Differentiability

- The limit concept is certainly indispensable for the development of analysis, for convergence and divergence of infinite series also depends on this concept.
- Theory of limits and then defining continuity, differentiability and the definite integral in terms of the limit concept is successfully executed by mathematicians.

## What Are Limits?

- Limit of a function may be a finite or an infinite number.
- If it just implies that the function f (x) tends to assume extremely large positive values in the vicinity of i.e..

- A function is said to be indeterminate at any point if it acquires one of the following values at that particular point
- The form is the standard indeterminate form.
- The point âââ cannot be plotted on the paper. It is just a symbol and not a number.
- Infinity (â) does not obey the laws of elementary algebra.

## Existence of Limit

The limit will exist if the following conditions get fulfilled:

Both LHS and RHS should be finite

## Some Important Limits

## Continuity

- A
**continuous function**is a function for which small changes in the input results in small changes in the output. Otherwise, a function is said to be discontinuous. - A function f (x) is said to be continuous at x = a if
- i.e.. L. H. L = R. H. L = value of the function at x = a
- A function f (x) is said to be
**discontinuous Function**.

## Differentiability

### Existence of Derivative

- Right and left hand derivative
- A piecewise function is differentiable at a point if both of the pieces have derivatives at that point, and the derivatives are equal at that point.
- In this case, Sal took the derivatives of each piece: first he took the derivative of at x = 3 and saw that the derivative there is 6.
- A function f is said to be continuously differentiable if the derivative fâ˛ (x) exists and is itself a continuous function.
- Though the derivative of a differentiable function never has a jump discontinuity, it is possible for the derivative to have an essential discontinuity.
- For example, the function

- is differentiable at 0,

### Differentiable Function

- A differentiable function of one real variable is a function whose derivative exists at each point in its domain.
- The graph of a differentiable function has a non-vertical tangent line at each interior point in its domain.
- A differentiable function is smooth (the function is locally well approximated as a linear function at each interior point) and does not contain any break, angle, or cusp.

### How Can a Function Fail to be Differentiable?

The function f (x) is said to be non-differentiable at x = a if

- Both R. H. D & L. H. D exist but not equal
- Either or both R. H. D & L. H. D are not finite
- Either or both R. H. D & L. H. D do not exist.