1.5: Limits We will learn about: limits, finding limits, one-sided limits, and unbound behavior of functions

Limits The word “limit” is used in everyday conversation to describe the ultimate behavior of something, as in the “limit of one’s endurance” or the “limit of one’s patience.” In mathematics, the word “limit” has a similar but more precise meaning.

Limits Given a function f(x), if x approaching 3 causes the function to take values approaching (or equaling) some particular number, such as 10, then we will call 10 the limit of the function and write

How to find limits? Graphically – look at the picture/graph Numerically – analyze the table of values Analytically – algebraic techniques

1. Graphically Find the limit: Draw graph: What is happening to y as x gets closer to 1?

2. Numerically This table shows what f (x) is doing as x approaches 3. Or, we have the limit of the function as x approaches 3. We write this procedure with the following notation. We write: 10 General notation: Or, as x → c, then f (x) → L If the functional value of f (x) is close to the single real number L whenever x is close to, but not equal to, c (on either side of c). x 2 2.9 2.99 2.999 3 3.001 3.01 3.1 4 f (x) 8 9.8 9.98 9.998 ? 10.002 10.02 10.2 12

Practice: Find the limit:

Things to Note: Looking at what happens to f (x) when x approaches c. It doesn’t matter what happens AT c, just what happens as x gets closer to c. x must approach c from either side of c. If the value when x approaches c from the left does not equal the value when x approaches c from the right, then… The Limit Does Not Exist!!!!

Definition If f (x) becomes arbitrarily close to a single number L as x approaches c from either side, then This is read as “the limit of f (x) as x approaches c is L.”

Properties of Limits Let b and c be real numbers, and let n be a positive integer.

Operations with Limits Let b and c be real numbers, let n be a positive integer, and let f and g be functions with the following limits.

Practice – Evaluate Analytically = -85 = -2 = 4

What happens now? is an indeterminate form. Use the Replacement Theorem: Let c be a real number and let f (x) = g (x) for all If the limit of g (x) exists as x approaches c, then the limit of f (x) also exists and

Factor and Cancel Practice – Pg. 93 # 42, 44, 46, 48

Rationalize the Numerator Practice – Pg. 91 # 5, 6 – algebraically

Practice:

One-Sided Limits While evaluating limits from graphs or tables: We saw: the numbers 2.9, 2.99, 2.999, ... approach 3 from the left, which we denote by x→3–, and the numbers 3.1, 3.01, 3.001, ... approach 3 from the right, denoted by x→3+. Such limits are called one-sided limits. x 2 2.9 2.99 2.999 3 3.001 3.01 3.1 4 f (x) 8 9.8 9.98 9.998 ? 10.002 10.02 10.2 12

One-Sided Limits Limit from the Left Limit from the Right

Practice! Evaluate the following = -1 = 1 Practice – Pg. 93 # 52, 54

Existence of a Limit If f (x) is a function and c and L are real numbers, then if and only if both the left and right limits exist and are equal to L

Unbounded Behavior Another way for the limit to fail to exist:

Function is unbounded – Limit does not exist Practice – Pg. 93 # 62, 64