Transcript
City University Of Science & Technology 1
City University Of Science & Technology
Limits & Continuity
Presented By:Haroon rasheedImran khan Ahmad yousafFahad noumanHasham zahid
Instructor:
Sir Nadeem Ahmad Sheikh 2
City University Of Science & Technology
Layout of Presentation
Limits • •
Properties of limits• •
One sided limits• Continuity
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Limits DefinitionIf f (x) is function of x and c, L are the real number, then
L is the limit of a function f (x) as x approaches c: (x) = L
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Example + 3x 7 )Solution: Apply the limits = + 3(-2) 7 = 4 6 7 = 4 13 = 9 Ans
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Uses of Limits in Daily Life
Reaction of two Compounds
Conversion of Ice to Water etc.
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Properties of limits
If = L & = MSum rule [ f (x) + g (x) ] = L + MQ: + 5x + 7
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One-Sided Limit Right Hand LimitLeft Hand Limit
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One-Sided Limit Right Hand Limit If x approach to “ a “ through value of x greater then “ a “ we say that x approach through the right and written asX a+0 or X
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Left Hand Limit If x approach to “ a “ through value of x less then “ a “ we say that x approach through the left and written asX a 0 or X
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Left Hand Limit Right Hand Limit
If L.H.L = R.H.L
Then exist
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Example Find f(x) if f (x) = x+1 ; x<=2 = 2x – 3 ; x > 2Sol: L.H.L f(x) = ( X + 1 ) = 2 + 1 = 3R.H.L f(x) = ( 2x - 3 ) = 2(2) - 3 = 1 f(x) f(x) f (x) does not exist
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Examplef (x) = |x|/x at x = 1 = -1
= 1
The left and right limits are different, therefore limit does not exist..
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Continuity Definition Example Properties
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Definition
A function f is continuous at a point x = c if 1. f (c) is defined
2.
3.
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x clim f (x) exists
x clim f (x) f (c)
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Examplef (x) = x – 1 at x = 2.
i) f(2) = 1ii) iii)
The limit exist! Therefore the function is continuous at x = 2.
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x 2lim x 1 1
x 2f (2) 1 lim x 1
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Check Continuity & Discontinuity
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By definition of g g(2) = 3 g(x)=x2 -4/(x-2)
limx → 2 g(x) = limx → 2 (x2 - 4)/(x - 2) =limx → 2 ( x- 2)(x + 2)/(x-2)
Now putting the limiting value = limx → 2 (x + 2) = 4
g(x) is discontinuous because limx → 2 g(x) ≠ g(2)
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Continuity Properties If two functions are continuous on the same interval, then their sum, difference, product, and quotient are continuous on the same interval..Every polynomial function is continuous..Every rational function is continuous.. 18
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ANY QUESTION ???
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