The Limit of a Function Section 2.2
The Limit of a Function To find the tangent to a curve or the velocity of an object, we now turn our attention to limits in general and numerical and graphical methods for computing them. Let’s investigate the behavior of the function f defined by f (x) = x2 – x + 2 for values of x approaches 2.
The Limit of a Function x f(x) 2.00 1 1.9 3.71 1.99 3.97 1.999 3.99 f (x) for values of x close to 2 but not equal to 2. x f(x) 2.00 1 1.9 3.71 1.99 3.97 1.999 3.99 2.001 4.003 2.01 4.03 2.1 4.31 3 8.00
The Limit of a Function What does this mean? f (x) L as x a
Example Guess the value of Solution: f (x) not defined when x = 1 So find limx1 f (x) Look at the table. x f(x) 2.00 .9 .99 3.71 .999 3.97 1.001 3.99 1.01 4.003 1.1 4.03
Example lim 𝑥→0 sin 𝑥 𝑥 Guess the value of Solution: f (x) not defined when x = 0 So find limx0 f (x) Look at the table. x f(x) .841 1 .1 .998 .01 .999 .001 .999 -.001 .999 -.01 .999 -.1 .998 -1 .841
One-Sided Limits The left-hand limit of f (x) as x approaches a is equal to L The right-hand limit of f (x) as x approaches a is equal to L
One-Sided Limits The function H is defined by H (t) approaches 0 as t approaches 0 from the left H (t) approaches 1 as t approaches 0 from the right. We indicate this situation symbolically by writing and
Example Use the graph of a function g to state the values (if they exist) of the following:
Example – Solution (c) Since the left and right limits are different, conclude from that limx2 g(x) does not exist. (f) This time the left and right limits are the same and so,
Infinite Limits limxa f (x) = is f (x) as x a
Infinite Limits
Infinite Limits
Example Find the vertical asymptotes of f (x) = tan x. Solution: Possible vertical asymptotes at cos x = 0 cos x = 0 cos-1 (0) = x x = /2
Example – Solution This shows that the line x = /2 is a vertical asymptote. cos x 0+ as x ( /2)– cos x 0– as x ( /2)+ y = tan x
2.2 The Limit of a Function Summarize Notes Read Section 2.2 Homework Pg.96 #1-4,7,9,11,15,19,27,29,31,41,45