Introduction to limits | Limits | Differential Calculus | Khan Academy

Defining a Simple Function and its Undefined Point00:00:00

The video begins by introducing the concept of a limit, a fundamental idea in calculus. It starts with a simple function, f(x) = (x-1)/(x-1). While seemingly equal to 1, this function is undefined when x = 1 because it results in 0/0, an indeterminate form. Thus, f(x) = 1 only if x is not equal to 1.

Graphing the Function and Understanding Undefined Points00:01:48

The speaker graphs the function f(x) = (x-1)/(x-1). It appears as a horizontal line at y=1, but with a gap (a hole) at x=1, indicating that the function is undefined at that specific point. This visual representation emphasizes that f(1) is undefined, even though the surrounding values are all 1.

Introduction to the Idea of a Limit00:03:25

The core idea of a limit is introduced by asking what the function is 'approaching' as x gets closer and closer to 1, rather than what its value is exactly at 1. In this example, as x approaches 1 from either side, the function's value approaches 1. This is written as: lim (x->1) f(x) = 1. The notation might seem complex, but the idea is simple: what value does the function get infinitesimally close to?

A More Complex Function with Discontinuity00:04:48

A second, more complex function, g(x), is introduced: g(x) = x^2 when x is not equal to 2, and g(x) = 1 when x is equal to 2. This function has a discontinuity, meaning its graph is not continuous.

Graphing the Discontinuous Function00:05:32

The speaker graphs g(x). It looks like a parabola (y=x^2) for all x except x=2. At x=2, there's a gap (a hole) where the parabola would normally be at y=4, and instead, a single point exists at (2,1).

Evaluating the Function vs. Finding the Limit00:08:05

The video highlights the difference between evaluating the function and finding its limit. For g(x), g(2) is defined as 1. However, the question of the limit as x approaches 2 of g(x) asks what value the function is getting close to as x approaches 2. By looking at the graph, as x gets closer to 2 from either side, the function's value (following the parabola) approaches 4, even though g(2) is 1. So, lim (x->2) g(x) = 4.

Numerical Approach to Understanding Limits00:09:29

To further illustrate the limit, the speaker uses a calculator to numerically evaluate g(x) (which is x^2 in this case) for values of x very close to 2. By plugging in values like 1.9, 1.99, 1.999, and also 2.1, 2.01, it's shown that the function's output gets progressively closer to 4, reinforcing the idea that the limit is 4, despite the function's actual value at x=2 being 1.