Lecture 4 - 1 CC2053: Calculus 1 Introduction to Calculus...

Info iconThis preview shows pages 1–14. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

Page1 / 40

Lecture 4 - 1 CC2053: Calculus 1 Introduction to Calculus...

This preview shows document pages 1 - 14. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online