What Are Limits?

Limits are the first building block of calculus. They help us understand what happens to a function as we get closer and closer to a certain point, but don't necessarily reach it. Imagine approaching a stoplight that turns yellow—you're almost there, but not quite!

Why Do We Need Limits?

Limits help us deal with situations where plugging in a value might not make sense, like dividing by zero. They allow us to describe instantaneous changes, which is super important in physics and engineering.

How Do We Write Limits?

The limit of \( f(x) \) as \( x \) approaches \( a \) is written as:

\[ \lim_{x \to a} f(x) \]

Key Ideas

• If \( f(x) \) gets closer to a number \( L \) as \( x \) gets closer to \( a \), then the limit is \( L \).
• If values don’t settle on a number, the limit may not exist.

One-Sided Limits

Sometimes we only care about approaching from one direction. For instance, \( x \to a^+ \) means from the right, and \( x \to a^- \) means from the left.

Tips for Evaluating Limits

• Plug in the number if possible.
• Try simplifying the function.
• Draw a graph!