An Introduction to Limits Objective: To understand the concept of a limit and To determine the limit from a graph.

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Presentation transcript:

An Introduction to Limits Objective: To understand the concept of a limit and To determine the limit from a graph

Calculus centers around 2 fundamental problems – 1)The tangent line- differential calculus P Q Instantaneous rate of change (Slope at a point) (Slope between 2 points)

2) The area problem- integral calculus Uses rectangles to approximate the area under a curve.

Limits: Yes – finally some calculus! Objective: To understand the definition of a limit and to graphically determine the left and right limits and to algebraically determine the value of a limit. If the function f(x) becomes arbitrarily close to a single number L (a y-value) as x approaches c from either side, then lim x c f(x) = L. *A limit is looking for the height of a curve at some x = c. *L must be a fixed, finite number. One-Sided Limits: lim x c+ f(x) =L 1 Height of the curve approach x = c from the right lim x c- f(x) =L 2 Height of the curve approach x = c from the left

Definition of Limit: If lim x c+ f(x) = lim x c- f(x) = L then, lim x c f(x)=L (Again, L must be a fixed, finite number.) f(2) = f(4) = Examples:

f(4) = f(0) = f(6) = f(3) =

Basic Limits (for the book part) lim x 4 2x – 5 = lim x -3 x 2 = lim x cos x = lim x 1 sin =

Important things to note: The limit of a function at x = c does not depend on the value of f(c). The limit only exists when the limit from the right equals the limit from the left and the value is a FIXED, FINITE #! A common limit you need to memorize: (see proof page 65 ) Limits fail to exist: (ask for pictures) 1. Unbounded behavior – not finite 2. Oscillating behavior – not fixed 3. - fails def of limit Do you understand how to graphically find a limit? HW: (odd) Use your calculator!