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Unformatted text preview: 1 CC2053: Calculus 1 Introduction to Calculus and Linear Algebra 2011/2012 Semester 1 CALCULUS: Limits, Continuity and Derivatives 2 CC2053: Calculus 1 Limits  The concept of limit What happens to the function as x approaches 1 ? ⇒ f ( x ) is not defined at x =1 1 2 ) ( 2 + = x x x x f 3 CC2053: Calculus 1 Limits Evaluate f ( x ) using values of x that get closer and closer to 1 from both the left and the right. x 0.8 0.9 0.95 0.99 0.999 1 1.001 1.01 1.05 1.1 f ( x ) 2.8 2.9 2.95 2.99 2.999 3.001 3.01 3.05 3.10 x approaches 1 from the right x approaches 1 from the left b × 4 CC2053: Calculus 1 Limits We say that “ the limit of f ( x ) as x approaches 1 equals 3 ” , or, 3 ) ( lim 1 = → x f x 5 CC2053: Calculus 1 Limits – Definition If f ( x ) gets closer and closer to a number L as x gets closer and closer to c from either side, then L is the limit of f ( x ) as x approaches c, or L x f c x = → ) ( lim 6 CC2053: Calculus 1 Limits Geometrically, means that the height of the graph of y = f ( x ) approaches L as x approaches c . L x f c x = → ) ( lim 7 CC2053: Calculus 1 Limits – Example 1 Geometric interpretation of the limit statement 3 1 2 lim 2 1 = + → x x x x 8 CC2053: Calculus 1 Limits Limits describe the behaviour of a function near a particular point, not necessarily at the point itself. L c f = ) ( L x f c x = → ) ( lim L x f c x = → ) ( lim L x f c x = → ) ( lim L c f ≠ ) ( defined not is ) ( c f M 9 CC2053: Calculus 1 Limits Two functions for which does not exist. ) ( lim x f c x → 10 CC2053: Calculus 1 Limits – Algebraic Properties of Limits If and exist, then ) ( lim x f c x → ) ( lim x g c x → ) ( lim ) ( lim )] ( ) ( [ lim x g x f x g x f c x c x c x → → → + = + ) ( lim ) ( lim )] ( ) ( [ lim x g x f x g x f c x c x c x → → → = ) ( lim )] ( [ lim x f k x kf c x c x → → = 1. Sum: 2. Difference: 3. Multiple: for any constant k ) ( lim ) ( lim ) ( ) ( lim x g x f x g x f c x c x c x → → → = P c x P c x x f x f )] ( lim [ )] ( [ lim → → = )] ( [lim )] ( [lim )] ( ) ( [ lim x g x f x g x f c x c x c x → → → = ) ( lim ≠ → x g c x 4. Product: 5. Quotient: 6. Power: if if exists P c x x f )] ( lim [ → 11 CC2053: Calculus 1 Limits – Limits of two linear functions For any constant k , and k k c x = → lim c x c x = → lim 12 CC2053: Calculus 1 Limits – Example 2 Find Solution: ) 3 2 ( lim 2 1 + → x x x ( ) ( ) 2 3 ) 1 ( 2 ) 1 ( 3 lim lim 2 lim ) 3 2 ( lim 2 1 1 2 1 2 1 = + = + = + → → → → x x x x x x x x 13 CC2053: Calculus 1 Limits – Example 3...
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 Spring '97
 KEITHCHAN

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