Core Calculus Concepts
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📍 What is a Limit?
A limit describes what value a function approaches as the input approaches a certain point.
Intuitive Definition: "What height is the function trying to reach as x gets closer and closer to a?"
limx→a f(x) = L
This means: As x gets arbitrarily close to 'a', f(x) gets arbitrarily close to 'L'.
Why Limits Matter
1. They handle "holes": Even if f(a) is undefined, the limit can still exist.
2. They define derivatives: The derivative is literally a limit of slopes.
3. They define integrals: Definite integrals are limits of Riemann sums.
4. They describe behavior: What happens as x → ∞? Limits tell us.
Common Limit Scenarios
Direct Substitution
If f(a) exists and is continuous, just plug in: limx→a f(x) = f(a)
Indeterminate Forms
0/0, ∞/∞ require algebraic tricks or L'Hôpital's rule
One-Sided Limits
Approaching from left (x→a⁻) or right (x→a⁺) separately
Limits at Infinity
What happens as x→∞ or x→-∞? Describes end behavior
🎓 Key Insight
Limits are about nearby values, not the value AT the point. Even if f(5) = 100, limx→5 f(x) could be 7 if the function "jumps" at x=5.
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