Discovering the Boundaries

Limits help us explore what happens to a function as the input gets super close to a certain value, without necessarily landing on it. Continuity is about making sure the function doesn't have any "jumps" or "holes".

Why Limits Matter

Calculus starts with limits because they let us talk about change and motion. Think of walking toward a door—you get closer, but what happens right as you reach it? Limits answer questions like, "What value does \( f(x) \) get closer to as \( x \) approaches 3?"

Continuity on a Graph

A function is continuous at a point if you can draw it there without lifting your pencil. If there’s a break, a hole, or a jump, it’s not continuous.

How to Find a Limit

• Plug in the value if possible.
• If you get something weird like \( \frac{0}{0} \), try to simplify.
• If there’s a hole, look at what value the function gets close to from both sides.

Real-Life Example

Predicting the speed of a car at an exact moment needs limits because speed is always changing.

Jumping into Continuity

If your water supply suddenly stops and then starts again, that’s a discontinuity! In math, we want functions to flow smoothly.