What Are Limits?

Limits are the foundation of calculus! They help us understand what happens to a function as the input (usually \( x \)) approaches a certain value. With limits, we can talk about values even when a function doesn't actually reach them.

Why Do We Need Limits?

Sometimes, plugging in a number into a function gives us something confusing—like dividing by zero. Limits let us peek at the function as we get super close to a tricky spot, without actually landing on it.

How Do We Find Limits?

• Direct Substitution: If plugging in works, that's your limit!
• Factoring: If direct substitution gives \( \frac{0}{0} \), try factoring and simplifying.
• Rationalization: For square roots, multiply by the conjugate.
• Left and Right Limits: Check what happens from both sides.

Common Limit Notation

• \( \lim_{x \to a} f(x) \) means "the limit of \( f(x) \) as \( x \) approaches \( a \)."

Why Are Limits Useful?

Limits tell us about continuity and are the key to understanding derivatives!