 It's useful to switch between the algebraic and geometric viewpoints. Algebra is based on numbers, expressions, and equations. For example, the limit is x approaches 3 of f of x equals 6 is an algebraic statement. Geometry is based on figures and shapes. For example, a graph is a geometric statement. For example, suppose we have the graph of y equals f of x, let's find f of 3 and the limit as x approaches 3 of f of x. So remember, equals means replaceable. Since this is a graph of y equals f of x, every question about f of x is a question about a y value. So f of 3 is the y value when x is equal to 3. So if we go to the point of the graph where x is equal to 3, we see that if x equals 3, y equals 2, equals means replaceable. So f of 3 is equal to 2. So the limit as x approaches 3 of f of x is the limit of the y values as x gets close to 3. Now an important idea here is, look both ways before you cross the street. Or rather, look both ways before you declare a limit. In this case, we can approach x equals 3 either from above or from below. And it's important to check out what happens in both cases. So as the x values get close to but stay more than 3, the y values get close to 2. And so we can say that the limit as x approaches 3 from above of f of x is equal to 2. Likewise, as the x values get close to but stay less than 3, the y values get close to 2. And so the limit as we approach 3 from below is 2. And since the limits agree, the limit as x approaches 3 of f of x is equal to 2. Or we might take another graph and let's find f of 2 and the limit as x approaches 2 of f of x. Again, equals means replaceable, and since this is graph of y equals f of x, we want to find the y value when x is equal to 2. So we go to the point, and the y value at x equals 2 is 3, so f of 2 is 3. So as we approach x equals 2 from above, the y values appear to be heading towards 3. So the limit as x approaches 2 from above is 3. But we approach x equals 2 from below, the y values appear to be heading towards 4. So the limit as x approaches 2 from below is 4. And since the limits disagree, the limit as x approaches 2 of f of x does not exist.