Evaluating Limits and One-Sided Limits

Evaluating Limits

• Simplifying and evaluating limits involves using techniques such as dividing out, rationalization, and substitution.
• The limit of a function can be evaluated by substitution, dividing out, or rationalization.

One-Sided Limits

• Right-hand limit: If the values of f(x) can be made close to L by taking values of x sufficiently close to a (but greater than a), then lim+ f(x) = L.

Unit Objectives

• After studying this unit, you should be able to define a limit, evaluate the limit of a function by substitution, dividing out, and rationalization, compute one-sided limits, and determine if the limit of a function exists.

Definition of Limit of Functions

• If the values of f(x) can be made as close as we like to a unique number L by taking values of x sufficiently close to a (but not equal to a), then lim f(x) = L.
• This means that the limit of the function f(x) as x approaches a is L.

Evaluating Limits Using Basic Limit Results

• Basic Limit Result: For any real number a and constant c, xlim x →a = a and xlim x →a c = c.
• Example: xlim x →−5 (3x + 2) = -17.

Computation of Limits by Substitution Method

• Evaluate lim x →−1 x2 + x − 6 / x + 3 = (-1)2 + (-1) - 6 / (-1) + 3 = 4 / 2 = 2.