Limits Fundamentals
Master the key concepts, notation, and evaluation techniques for limits.
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Questions Covered in This Set
8 cards to master
What does the notation lim(x→c) f(x) = L mean?
As x gets closer to c, f(x) gets closer to L. We care about what happens near c, not necessarily at c.
Can a limit exist even if the function is undefined at that point?
Yes! Limits describe behavior as you approach a point, not the value at that point. Example: lim(x→2) (x²-4)/(x-2) = 4, even though f(2) is undefined.
What are the three main approaches to evaluating limits?
1) Numerical: create a table of values approaching the target; 2) Graphical: plot and observe the y-value approached; 3) Algebraic: simplify through factoring or other techniques.
What is a one-sided limit?
A limit that approaches from only one direction. lim(x→c⁺) approaches from the right (positive), lim(x→c⁻) approaches from the left (negative).
When does a two-sided limit exist?
When both one-sided limits (left and right) exist AND are equal to each other.
How do you resolve a 0/0 indeterminate form?
Use algebraic manipulation such as factoring and canceling common terms, then evaluate the simplified expression.
For g(x) = |x|/x, why doesn't lim(x→0) g(x) exist?
Because lim(x→0⁺) g(x) = 1 and lim(x→0⁻) g(x) = -1. The left and right limits are different, so the two-sided limit does not exist.
What does a 'hole' in a graph indicate about limits?
The function is undefined at that point, but the limit exists. The graph approaches a specific y-value from both sides even though there's no point plotted there.
