Section 1 2 Finding Limits Graphically and Numerically

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. f(x) L x c The limit of f(x)… is L. Notation: as x approaches c…

Calculating Limits Our book focuses on three ways: 1. Numerical Approach – Construct a table This of values Lesson 2. Graphical Approach – Draw a graph 3. Analytic Approach – Use Algebra or calculus Next Lesson

Example 1 Use the graph and complete the table to find the limit (if it exists). x 1. 999 2 2. 001 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. 001 -1 -. 999 -. 9 f(x) -2. 1 -2. 001 DNE -1. 999 -1. 9 If the function is not continuous at the value of x, a graph and table can be very useful.

Example 3 Use the graph and complete the table to find the limit (if it exists). -6 x -4. 1 -4. 001 -4 -3. 999 -3. 9 f(x) 2. 999 -6 8 2. 999 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) 0 -1 1 -1 DNE 1 -1 1

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.