Welcome to the concept of limits! A limit is a fundamental idea in calculus that describes what value a function approaches as the input gets arbitrarily close to a specific point. Notice how even though the function has a hole at x equals 2, the limit still exists and equals 4.
The key idea of a limit is approaching. Watch as the red dot moves along the function, getting closer and closer to x equals 2. Even though there's a hole at that point, the function values approach 4. This approaching behavior from both the left and right sides is what defines the limit.
A limit is one of the fundamental concepts in calculus. It describes the behavior of a function as the input approaches a particular value. Even if the function isn't defined at that exact point, we can still determine what value it approaches. This concept allows us to analyze continuity, derivatives, and integrals.
Watch as we move the point along the function. As x approaches 2 from either side, the function value approaches 4. This is the essence of a limit - the approaching behavior matters more than what happens exactly at the point. The red dot shows how the function value changes as we get closer to our target.
The formal definition of a limit uses epsilon and delta. For any small distance epsilon around the limit value L, we can find a corresponding distance delta around the point a. This ensures that whenever x is within delta of a, the function value f(x) is within epsilon of L. Watch as we adjust these bands to show this relationship.
Not all limits are the same. We can have one-sided limits, where we approach from only the left or only the right. In this example, the left limit as x approaches 2 is 3, while the right limit is 4. Since these one-sided limits are different, the two-sided limit does not exist. For a limit to exist, both one-sided limits must exist and be equal.
Limits are the foundation of calculus. They define derivatives as the limit of secant line slopes as the distance approaches zero, creating tangent lines. They also define integrals as limits of Riemann sums. Watch as the secant line approaches the tangent line, demonstrating how limits create derivatives. Limits help us understand continuity, asymptotic behavior, and form the theoretical backbone of all advanced mathematics.