Section 1.2 - Finding Limits Graphically and Numerically

Limit Informal Definition: If f(x) becomes arbitrarily close to a single REAL number L as x approaches c from either side, the limit of f(x), as x appraches c, is L. c L f(x) x The limit of f(x)… is L. Notation: as x approaches c…

Calculating Limits Our book focuses on three ways: Numerical Approach – Construct a table of values Graphical Approach – Draw a graph Analytic Approach – Use Algebra or calculus This Lesson Next Lesson

Example 1 Use the graph and complete the table to find the limit (if it exists). x 1.9 1.99 1.999 2 2.001 2.01 2.1 f(x) 6.859 7.88 7.988 8 8.012 8.12 9.261 If the function is continuous at the value of x, the limit is easy to calculate.

Example 2 Use the graph and complete the table to find the limit (if it exists). Can’t divide by 0 x -1.1 -1.01 -1.001 -1 -.999 -.99 -.9 f(x) -2.1 -2.01 -2.001 DNE -1.999 -1.99 -1.9 If the function is not continuous at the value of x, a graph and table can be very useful.

The limit does not change if the value at -4 changes. Example 3 Use the graph and complete the table to find the limit (if it exists). -6 x -4.1 -4.01 -4.001 -4 -3.999 -3.99 -3.9 f(x) 2.9 2.99 2.999 -6 8 2.999 2.99 2.9 If the function is not continuous at the value of x, the important thing is what the output gets closer to as x approaches the value. The limit does not change if the value at -4 changes.

Three Limits that Fail to Exist f(x) approaches a different number from the right side of c than it approaches from the left side.

Three Limits that Fail to Exist f(x) increases or decreases without bound as x approaches c.

Three Limits that Fail to Exist f(x) oscillates between two fixed values as x approaches c. Closest Closer Close x f(x) -1 1 DNE

A Limit that DOES Exist If the domain is restricted (not infinite), the limit of f(x) exists as x approaches an endpoint of the domain.

Example 1 Given the function t defined by the graph, find the limits at right.

Example 2 Sketch a graph of the function with the following characteristics: does not exist, Domain: [-2,3), and Range: (1,5) does not exist, Domain: (-∞,-4)U(-4,∞), and Range: (-∞,∞)

Why is there a lot of “noise” over here? Classwork Sketch a graph and complete the table to find the limit (if it exists). x -0.1 -0.01 -0.001 0.001 0.01 0.1 f(x) 2.8680 2.732 2.7196 DNE 2.7169 2.7048 2.5937 This a very important value that we will investigate more in Chapter 5. It deals with natural logs. Why is there a lot of “noise” over here?