MATHEMATICS

RELATIONS AND FUNCTIONS

Limits, Continuity and Differentiability

Every Monday - Wednesday - Friday 6 pm (1 hour FREE Class)

Today: Continuity 2- Problem Solving

Monday: Derivability - Basics + Problem Solving

Graphical Interpretation of Continuity and Differentiability

MATHEMATICS

RELATIONS AND FUNCTIONS

Limits, Continuity and Differentiability

MATHEMATICS

RELATIONS AND FUNCTIONS

Limits, Continuity and Differentiability

MATHEMATICS

RELATIONS AND FUNCTIONS

Limits, Continuity and Differentiability

MATHEMATICS

RELATIONS AND FUNCTIONS

Limits, Continuity and Differentiability

MATHEMATICS

RELATIONS AND FUNCTIONS

Limits, Continuity and Differentiability

MATHEMATICS

RELATIONS AND FUNCTIONS

Limits, Continuity and Differentiability

MATHEMATICS

RELATIONS AND FUNCTIONS

Limits, Continuity and Differentiability

MATHEMATICS

RELATIONS AND FUNCTIONS

Limits, Continuity and Differentiability

MATHEMATICS

RELATIONS AND FUNCTIONS

Limits, Continuity and Differentiability

MATHEMATICS

RELATIONS AND FUNCTIONS

Limits, Continuity and Differentiability

MATHEMATICS

RELATIONS AND FUNCTIONS

Limits, Continuity and Differentiability

MATHEMATICS

Continuity and Dis-Continuity

f (x) = sin ( log |x| ), Check the continuity at x = 0

Example

MATHEMATICS

Continuity and Dis-Continuity

f (x) = sin ( log |x| )

Solution:At x = 0; lim f (0)x → 0– = Not defined

Not definedlim f (0)x → 0+ =and

It is essential discontinuity

, Check the continuity at x = 0

log |x| tends to – ∞

So

Limit doesn’t exist

So it is not continuous at x = 0

Example

sin (– ∞) =

sin (– ∞) =

MATHEMATICS

Continuity and Dis-Continuity

f (x) = Lim n → ∞

πx2sin

2n

Check the continuity at all values of x

Example

MATHEMATICS

Continuity and Dis-Continuity

f (x) = Lim n → ∞

πx2sin

2n

f (x) = Lim n → ∞

πx2sin

2n 1 ; x ∈ odd ; x ∉ odd

=0

1

–5 –4 –3 –2 –1 1 2 3 4 5

Solution:

So it is discontinuous at all x ∈ odd

Check the continuity at all values of x

It is removable discontinuity

Example

MATHEMATICS

Continuity and Dis-Continuity

If f (x) =log (x + 2) – x2n sin x

1 + x2n Check continuity at x = 1lim

n→∞

Example

MATHEMATICS

Continuity and Dis-Continuity

If f (x) =log (x + 2) – x2n sin x

1 + x2n Check continuity at x = 1

Solution

RHL| x=1 = limx→1

+limn→∞

log (x+2) – x2n sinx1 + x2n

= limx→1

+ limn→∞

log (x+2) – sin xx2n

+ 1x2n1

= –sin l

limn→∞

These values tend to 0

Example

MATHEMATICS

Continuity and Dis-Continuity

LHL| x=1 = limx→1

–limn→∞

log (x+2) – x2n sinx1+x2n

= log 3

LHL| x=1 ≠ RHL| x=1 but they exist !!

Jump discontinuity

These values tend to 0

Every Monday - Wednesday - Friday 6 pm (1 hour FREE Class)

Today: Continuity 2- Problem Solving

Monday: Derivability - Basics + Problem Solving

All the very best :-)DREAM ON!