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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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City University Of Science & Technology
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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City University Of Science & Technology
Example + 3x 7 )Solution: Apply the limits = + 3(-2) 7 = 4 6 7 = 4 13 = 9 Ans
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City University Of Science & Technology
Uses of Limits in Daily Life
Reaction of two Compounds
Conversion of Ice to Water etc.
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City University Of Science & Technology
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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City University Of Science & Technology
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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City University Of Science & Technology
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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City University Of Science & Technology
Left Hand Limit Right Hand Limit
If L.H.L = R.H.L
Then exist
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City University Of Science & Technology
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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City University Of Science & Technology
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
City University Of Science & Technology
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)
City University Of Science & Technology
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)
City University Of Science & Technology
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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City University Of Science & Technology
ANY QUESTION ???
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