MATHEMATIC S RELATIONS AND FUNCTIONS Limits, Continuity and Differentiability
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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!
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