Calculus Limits: Approaching the Impossible
A Limit defines how a function behaves as it gets closer and closer to a certain point. It is the concept that allows us to handle infinity and division by zero. Without limits, there is no calculus.
1. What Does "Approach" Mean?
Imagine you are walking halfway to a wall. Then you walk halfway again. Then halfway again. Will you ever touch the wall? Mathematically, no. But the limit of your position is the wall.
lim(x→c) f(x) = L
"As x gets closer to c, f(x) gets closer to L."
2. Evaluating Limits
- Direct Substitution: Just plug in the number. If you get a valid number, you're done. (e.g., lim x→2 of x+3 = 5).
- Factor & Cancel: If you get 0/0, try factoring. (x²-1)/(x-1) becomes (x+1)(x-1)/(x-1). The (x-1) cancels out, leaving (x+1). Now plug in x=1 to get 2.
- L'Hôpital's Rule: The ultimate weapon. If you get 0/0 or ∞/∞, take the derivative of the top and bottom separately.
3. Continuity
A function is Continuous if you can draw it without lifting your pencil. For a function to be continuous at point c:
- f(c) must exist (no holes).
- The limit as x→c must exist (left match right).
- The limit must equal f(c).
4. Solving "Does Not Exist" (DNE)
A limit does not exist if:
- The left side goes to 5 but the right side goes to 7 (Jump Discontinuity).
- The function shoots up to infinity (Vertical Asymptote).
- The function oscillates wildly (like sin(1/x) near 0).