Section 2.3 Calculating Limits Using the Limit Laws "It is our choices, Harry, that show what we truly are, far more than our abilities." - Albus Dumbledore

(The limit of a sum is the sum of the limits.) (The limit of a difference is the difference of the limits.) The limit of a constant times a function is the constant times the limit of a function. The limit of a product is the product of the limits. The limit of a quotient is the quotient of the limits (provided that the limit of the denominator is not 0).

where n is a positive integer where n is a positive integer where n is a positive integer where n is a positive integer (If n is even, we assume that a>0.) (If n is even, we assume that

Direct Substitution Property: If f is a polynomial or a rational function and a is in the domain of f, then:

Example 1

Example 2 Evaluate the following limits and justify each step:

Example 3 Find

If when , then , provided the limits exist.

Example 4 Find Where

Example 5 Evaluate

Example 6 (the trick!)

Theorem: If and only if And

Example 7 Show that

Example 8: Prove that does not exist.

Example 9 If Determine whether Exists.

Example 10 the largest integer that is less than or equal to x. (For instance, The greatest integer function is defined by Show that does not exist.