Introduction to Limits of a Function: Finding Limits using the Table of Values: Detailed Explanation

The video introduces the concept of limits by first evaluating a function f(x) = (x^2 - 1) / (x - 1) at x=1, showing it results in an undefined value (0/0) and a hole in the graph (discontinuity). This highlights that a limit is not about the function's value at a specific point, but what it approaches.

Instead of evaluating the function at a specific point, the video demonstrates how to find the limit by setting values of x that are increasingly closer to the target value (x=1 in the example), from both the left and the right. This involves creating a table of values to observe the function's behavior.

Using the function f(x) = (x^2 - 1) / (x - 1), the video shows how to select x-values approaching 1 from the left (e.g., 0.5, 0.9, 0.999) and from the right (e.g., 1.5, 1.1, 1.01). By evaluating f(x) for these values, it's observed that the function approaches 2 from both sides, indicating the limit is 2.

The video summarizes that as x approaches 1, the function's value approaches 2 from both the left and right sides, visually confirming the limit is 2. The standard notation for expressing a limit is also introduced.

A second example, finding the limit of (1 + 3x) as x approaches 2, is presented with a detailed five-step process: creating a table, setting x-values closer to 2, solving for f(x), comparing the approach from left and right, and concluding the limit. The function is shown to approach 7 from both sides.

The video demonstrates finding the limit of a piecewise function as x approaches 4. It shows that if the function approaches different values from the left and right sides, the limit does not exist (DNE). In this example, the limit approached from the left is 5, while from the right it's not and therefore the limit does not exist.

The video concludes by reiterating the five steps for finding the limit of a function using the table of values: create a table, set x values closer to the target, solve for f(x), compare, and conclude. It emphasizes that a limit exists only if the approaches from both sides are equal.