### Limit:

A function ‘

f‘ is said to have limit ‘L‘ if ‘f‘ is defined for larger value of ‘x‘ as approaches to infinity,f(x)approaches to ‘L‘.

i.e

For example if *f(x) = 1/x*

*then*

* f(1) = 1/1 = 1*

* f(10) = 1/10 = 0.1*

*f(100) = 1/100* = 0.01

*f(1000) = 1/1000 = 0.001*

as *x* approaches infinity *f(x)* approaches 0.

Collectively,

A function

f(x)is said to have a limit ‘L‘ iff(x)defined for all values of ‘x‘ close to ‘a‘ (need notx = a) and as approachesa

e.g

*f(x) = x^2 – 4 / x – 2* , *x =/= 2*

*f(x) = (x+2) (x-2) / (x-2)*

*f(x) = (x+2) , x =/= 2*

as *x* approaches to 2 , *f(x)* approaches to 4

### Rules for Finding Limits

#### Theorem 1 (Properties of Limits)

#### Theorem 2

e.g If *p(x) = 3x^2 + 2x + 1*

Then = 3(2)^2 + 2(2) + 1

=17

#### Theorem 3

If *P(x)* and *Q(x)* are polynomials and *Q(c)* =/= 0, then

e.g

#### Theorem 4 (Sandwich Theorem)

Suppose that *g(x) <= f(x) <= h(x)* for all *x* in open interval containing ‘*c*‘ except possibly at *x=c* itself.

Suppose also that

e.g *Cos x < Sin x / x < 1*

so by sandwich theorem

1-

### Theorem (One sided vs Two sided)

A function

f(x)has a limit as ‘x‘ approaches to ‘c‘ if and only if it has left hand limit and right hand limit atx=cand both are equal

### Continuity

#### Continuity of a function at a Point

**(Continuity Test)**

A function *f(x)* is said to be continuous at *x=c* , *c* element *Df* if and only if it meets the following three conditions

The example 2 given above is a perfect example of a continuous function.

A function ‘

f‘ is continuous in an interval[a,b]if it is continuous for allxelement[a,b]

#### Theorem 1 (Continuity of Algebraic combinations)

If functions ‘*f *‘ and ‘*g*‘ are continuous at *x=c* , the following functions are also continuous at *x=c *

(1) *f+g & f-g*

(2) *fg*

(3) *f/g, *provided* g(c) *=/= 0

(4) k*f * where ‘ k’ is any constant

(5) *[ f(x) ]^*m/n , provided *[ f(x) ]^*m/n is defined on an interval containing ‘c’ and m,n are integers

#### Theorem 2

Every polynomial is continuous at every point of real line. Every rational function is continuous at every point where it’s denominator is not zero.

f(x) = x+1 / x-1

continuous at every point except 1.

#### Theorem 3

If ‘*f *‘ is continuous at ‘*c *‘ and ‘*g*‘ is continuous at *f(c)* then *gof* is continuous at *x=c.*

### Types of disContinuity

#### Non-Removable discontinuity

If at any point a function does not have functional value as well as limiting value then the type of discontinuity is called non-removable discontinuity

e.g

f(x)=1/x

f(0) does not exist

#### Removable disContinuity

If at any point , a function does not have any functional value but it has limiting value at that point then that kind of discontinuity is called removable discontinuity

e.g

f(x) = x^2 – 4 / x-2

f(2) does not exist

#### Continuous Extension

If *f(c)* does not exist but then we can define a new function *f(x)* as

Then the function *f(x)* is continuous at *x=c* . It is called continuous extension of *f(x)* to *x=c*.

#### Example 1

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