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Calculus
Limits and Continuity
How Graphs Behave Near a Point
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Limits describe what value a function gets close to as the input approaches a particular number. They are essential because a graph can behave predictably near a point even if the function is not defined exactly at that point. This idea lets calculus handle holes, jumps, vertical asymptotes, and changing rates. Continuity builds on limits by asking whether the graph connects smoothly at the point with no break.
Key Facts
- lim x -> a f(x) = L means f(x) gets arbitrarily close to L as x gets close to a.
- The left-hand limit is lim x -> a- f(x), and the right-hand limit is lim x -> a+ f(x).
- A two-sided limit exists only if lim x -> a- f(x) = lim x -> a+ f(x).
- A function is continuous at x = a if f(a) is defined, lim x -> a f(x) exists, and lim x -> a f(x) = f(a).
- A removable discontinuity is a hole where the limit exists but f(a) is missing or has the wrong value.
- For polynomials and many basic functions, if the function is continuous at a, then lim x -> a f(x) = f(a).
Vocabulary
- Limit
- A limit is the value a function approaches as the input gets closer and closer to a chosen number.
- One-sided limit
- A one-sided limit describes what a function approaches from only the left side or only the right side of a point.
- Continuity
- Continuity at a point means the function value matches the limit there and the graph has no break at that point.
- Removable discontinuity
- A removable discontinuity is a hole in the graph where the limit exists but the function value is missing or different.
- Jump discontinuity
- A jump discontinuity occurs when the left-hand and right-hand limits approach different values.
Common Mistakes to Avoid
- Substituting into every limit immediately, which is wrong when the function has a hole, jump, or undefined expression at the point. First check the graph or simplify the expression to see what the function approaches.
- Confusing f(a) with lim x -> a f(x), which is wrong because the value at the point can differ from the value approached nearby. A filled dot and an open circle may represent different y-values.
- Saying a two-sided limit exists when only one side approaches a value, which is wrong because both one-sided limits must exist and be equal. Always compare the left-hand and right-hand behavior.
- Calling a graph continuous just because it has a defined point, which is wrong because the function value must also match the limit. A single filled dot away from the curve does not fix a hole in the curve.
Practice Questions
- 1 For f(x) = (x^2 - 9)/(x - 3), find lim x -> 3 f(x) and state whether f is continuous at x = 3.
- 2 A graph has lim x -> 2- f(x) = 5, lim x -> 2+ f(x) = 1, and f(2) = 5. Does lim x -> 2 f(x) exist? Is the function continuous at x = 2?
- 3 A curve approaches the open point (4, 7) from both sides, but there is a filled point at (4, 3). Explain the limit, the function value, and the type of discontinuity.