An Introduction to Limits

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

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

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

6. 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 limx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: limxc+ f(x) =L1 Height of the curve approach x = c from the right  limxc- f(x) =L2 Height of the curve approach x = c from the left

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

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

9. Basic Limits (for the book part) • limx4 2x – 5 = • limx-3 x2 = • limxcos x = • limx1sin =

10. 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 63 ) • 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?Assign: WS and bookwork for 1.2