Every Monday - Wednesday - Friday 6 pm (1 hour FREE Class)
Today: Continuity 2- Problem Solving
Monday: Derivability - Basics + Problem Solving
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