Lecture, 01/27 Ex: Sketch the sign table for the rational function f(x) = (x3 (x – 1)4) / (x – 3)2 Roots: 0, 1, 3 - + + + Even number of 0’s = no sign change Odd number of 0’s = sign change As we approach 3, it becomes unbounded because the numerator is finite and the denominator is 0. We say that the line x = 3 is the vertical asymptote. Ex: Find the domain of the following functions: a) x3/2 = (x3 so x ≥ 0 so D = [0,() b) x-2/3 = 1/(x so x ≠ 0 so D = (-(,0) u (0,() c) x-3/4 = 1/4(x3 so x ≠ 0 and x3 ≥ 0 and therefore x ≥ 0 together x ( 0 so D = (0,() Limits Ex: Sketch the graph of f(x) = (x2 – x – 2) / (x – 2) x = 2 is not part of the domain otherwise let x≠ 2 f(x) = ((x – 2) (x + 1)) / (x – 2) and the (x – 2)’s cancel = x – 1 Graph: Rmk: We cannot figure the y-coordinate of the hole from the function value, instead we approach from outside x = 2, from the left or the right to figure out the value. We will denote this situation by: Rmk: The hole at x = 2 is so small that id we sketch the real picture of this function, the hole is not visible. Dfn: We say that a function f(x) has limit L at x = a if f(x) approaches a, but never touches a and we can get close to L as much as we want approaching on: Or: f(x) Or: f(x) Rmk: The limit L is independent from the value of the function f(a), it may or may not be equal to f(a), and f(a) might not even exist as in the example. Ex: Find If we plug in x = 0 we get 0/0 which does not give us any information Try cancellations: x ≠ 0 (12 + 2x + x2 – 1) / x = (2x + x2) / x = (x (2 + x )) / x = 2 + x So 2 + 0 = 2 is supposed to be the limit. Rmk: We call this situation an indeterminacy. This may give different results Properties of Limits Let 1. *If c is a constant* 2. 3. If M ≠ 0, then 4. Ex: Rmk: The limit might not exist as a number, sometimes we denote this as DNE. Ex: Find the If x approaches 1 when x approaches 0 from the right, and if x approaches -1 when x approaches 0 from the left, then the limit DNE because 1 and -1 are not the same so it has no limit at 0. DNE. Dfn: Right-hand and left-hand limits are defined similarly. We write: if f(x) approaches L as x approaches a from the right if f(x) approaches L as x approaches a from the left The limit of x exists if as numbers In the above example: