Evaluating Limits: Tables of Values

Evaluating Limits Through Table of Values

• The lesson is about evaluating limits by using tables of values.
• The concept of limits is introduced and defined. The lesson illustrates how to use tables of values to estimate the limit of a function.
• One learning objective is to illustrate the limit of a function using a table of values.
• The lesson introduces left-hand and right-hand limits.
• Evaluating limits via tables of values is also a means to illustrate the concept of infinite limits.
• A limit of a function is defined as taking the value of a function as the value of the independent variable gets arbitrarily close to a target value.
• One-sided limits—left-hand and right-hand limits—are used to evaluate limits where the function approaches a specific value from one side of a point.
• It is important to understand that a function does not need to be defined at the point a in order for the limit at that point to exist.
• The limit of a function at a point can only be determined through examination of the function's behavior when the input is arbitrarily close to the target value, and thus limits give a better sense of function behaviors than specific-point values.
• Limits play a critical role in areas like calculus, enabling us to study how a function behaves at a specific point or when inputs are close to a point.

Limit of a Function

• The example focuses on the function f(x) = x + 4, investigating how the values of the function change as x approaches 2.
• Tables illustrate values of f(x) around 2. -The left-hand limit of the function as x approaches 2 is 6. -The right-hand limit of the function as x approaches 2 is 6.
• The value of the function f(x) approaches 6 as x approaches 2. Thus the limit is 6.

Intuitive Definition of a Limit

• The function f(x) is defined when x is close to c.
• If f(x) gets closer and closer to L as x gets infinitely closer to c, from both sides, the limit of f(x) as x approaches c is equal to L.

One-Sided Limits

• Left-hand Limit: If the function f(x) is defined when x is near c from the left, then the limit of f(x) as x approaches c from the left is equal to a number M.
• Right-hand Limit: If the function f(x) is defined near c from the right, then the limit of f(x) as x approaches c from the right is equal to a number N.
• If the left-hand and right-hand limits are the same, then the two-sided limit exists and is equal to their common value.

Infinite Limits

• A function f(x) may not have a limit at a certain value (c) if it increases or decreases indefinitely as x approaches c.
• If f(x) increases without bound as x approaches c, then the limit is infinity.
• If f(x) decreases without bound as x approaches c, then the limit is negative infinity.

Practice problems involving limit estimations

• Limit estimation problems using tables of values are included for the functions g(x) = –x² – 6x + 14, g(x)= 5x + 8, and g(x) = √x, focusing on applying the concept to specific examples.
• Limit estimation practice also involves the signum function s(x).
• Practice exercises using piecewise functions such as t(x) are included, examining specific cases around x = 1.

Further practice

• Practice estimation of limits using specific equations in the form (x^2 – 2x + 6) , demonstrating application of limit findings.
• Estimating limits to determine limit of r(x) using a provided table.

Additional Practice exercises

• Additional problems involving functions and their limits are outlined for the interested student.
• The practice problems presented include a selection focusing on the use of tables to estimate limits.

Summary of Limit Evaluation

• To estimate the limit of a function f(x) as x approaches c, follow these steps:
  1. Construct tables with x-values near c from both the left and right.
  2. Determine the corresponding f(x) values for those x-values.
  3. Evaluate the values that f(x) is approaching in both tables from the left and the right of c. If these values are the same, the limit exists and is equal to these common values. If they're different the limit does not exist.