Grasping the Idea of Limits

Limits are the foundation of calculus! They help us understand what happens to a function as we get super close to a specific point—even if we never actually reach it. Imagine walking towards a wall: with every step, you halve the distance to the wall, but never quite touch it. Limits are how math describes this phenomenon.

Continuity: No Sudden Jumps

A function is called continuous if you can draw it without lifting your pencil from the paper. This means there are no breaks, jumps, or holes in its graph. If the limit from the left and right at a point matches the function's value there, the function is continuous at that point.

Why Are Limits and Continuity Important?

These concepts let us rigorously define derivatives and integrals, the main tools of calculus. They also help us analyze real-world scenarios where things change smoothly or suddenly.

How to Find a Limit

• Plug in the value directly if possible.
• If you get an indeterminate form like \( \frac{0}{0} \), try simplifying or factoring.
• Use special techniques like L'Hôpital's Rule (later!).