Evaluating Limits and One-Sided Limits

What is the value of lim (x → -5) (3x + 2)?

What is the value of lim (x → 2) (2x^2 - 3x + 1)?

What is the value of lim (x → -1) (x^2 + x - 6)/(x + 3)?

Which of the following is a limit law?

What is the value of lim (x → 2) (x^2 + 4)?

Which of the following is an example of one-sided limit?

What is the value of lim (x → -5) (3x)?

What is the value of lim (x → 2) (x - 2)?

What is the value of lim (x → -1) (x + 3)?

Which of the following is a step in evaluating limits using the substitution method?

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.

Learn about simplifying and evaluating limits using techniques like dividing out, rationalization, and substitution. Understand one-sided limits and how to define a limit.