Before you can do Calculus (Derivatives and Integrals), you must understand Limits. Limits are the mathematical tool we use to handle the dangerous stuff: dividing by zero, going to infinity, and adding up infinite items.

A Limit asks: "Where is the graph GOING, even if it never actually gets there?"

1. The Hole in the Road Analogy

Imagine you are driving down a road, and there is a bridge out at mile marker x=5. You can drive to 4.9. You can drive to 4.99. You can drive to 4.999.
Even though the bridge doesn't exist at exactly 5, we can say "The limitation of your path is mile 5". Math is the same way.

2. The Most Famous Limit: 0/0

If you plug in x=1 into the function (x²-1)/(x-1), you get 0/0.
In Algebra, you are told this is "Undefined".
In Calculus, we call this Indeterminate. It doesn't mean "No Answer"; it means "Work Harder".

By factoring the top: (x-1)(x+1) / (x-1), the indeterminate parts cancel out! You are left with (x+1).
So as x approaches 1, the answer approaches 2. The "Hole" has a location.

3. One-Sided Limits (Left vs Right)

Sometimes a graph is broken.
• Limit from Left (x -> a⁻): What height do you approach coming from negative infinity?
• Limit from Right (x -> a⁺): What height do you approach coming from positive infinity?

Rule: The General Limit exists IF AND ONLY IF the Left Limit equals the Right Limit. If they point to different numbers (like a Jump Discontinuity), the Limit Does Not Exist (DNE).

4. L'Hôpital's Rule: The Cheat Code

If you plug in a limit and get 0/0 or ∞/∞, you don't need to factor. You can use the Derivative.

Example: Limit x->0 of sin(x)/x. Plug in 0, get 0/0.
Derivative of sin(x) is cos(x). Derivative of x is 1.
New Limit: cos(0)/1 = 1/1 = 1. Solved instantly!

5. Limits at Infinity

What happens if x gets REALLY big? This is asking for the Horizontal Asymptote.

• Top Heavy (x³ / x²): Goes to Infinity.
• Bottom Heavy (x² / x³): Goes to Zero.
• Balanced (3x² / 5x²): Goes to the ratio of coefficients (3/5).

6. Continuity

Calculus requires "Continuous Functions". A function is continuous at point C if three things are true:
1. f(c) exists (Real point)
2. Limit x->c exists (Left = Right)
3. The Limit = The Point.

This seems formal, but it ensures knowing that the bridge is not out and the road connects smoothly.

7. FAQ

Q: Can a limit equal infinity?
A: Yes. If the graph shoots up forever (like 1/x² at x=0), we write Limit = ∞. This is a specific type of "Does Not Exist".

Q: Is 0.999... equal to 1?
A: Yes! Using limits of infinite geometric series, we can prove that 0.999... is exactly equal to 1, not just close.

Conclusion

Limits are the foundation of Analysis. They allow us to tame the concept of Infinity and harness it to solve practical problems in motion, growth, and physics.