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.
Understanding Limits and Continuity
A limit studies nearby inputs, not the value produced at the target input. This distinction is useful when an expression has a factor that makes it fail at one number. For example, a fraction made from x squared minus four over x minus two cannot be evaluated when x is two.
Factoring the top gives x minus two times x plus two. Away from two, the fraction behaves exactly like x plus two. Numbers such as one point nine and two point zero one produce outputs close to four.
The missing point does not change the nearby pattern. This is why algebra can reveal information that a direct substitution hides.
Approach from each side matters because a graph can follow different rules before and after one input. Piecewise functions often do this. A jump in a taxi fare, a tax bracket, or a shipping charge can be modeled with separate rules.
At the boundary, the output from smaller inputs may differ from the output from larger inputs. Looking at only one side can give an incomplete answer. Vertical asymptotes need similar care.
As inputs move toward an asymptote, outputs may grow larger in magnitude without settling near one ordinary number. A calculator screen may show very large values, but that does not mean the function reaches infinity as an output.
Continuity is important because many models assume small changes in input create small changes in output. Temperature at a fixed place usually changes continuously over short periods, though a thermometer records rounded values. The height of a moving object is continuous while it stays in motion.
A light switch is better modeled by a jump, since its state changes abruptly. In calculus, continuity supports later ideas about rates of change and accumulated change. A function that has a derivative at a point must be continuous there.
The reverse is not always true. A graph can be unbroken yet have a sharp corner, where it has no single instantaneous slope.
When solving limit problems, first identify the kind of expression and the source of difficulty. Direct substitution works for many familiar functions when the input lies in their domain. If substitution produces an undefined fraction, factoring may cancel a common factor.
Rationalizing can help when square roots are involved. For a piecewise rule, calculate the behavior from each side using the appropriate rule. Graphs and value tables are useful checks, but they are not final proof.
A graph may hide a hole because of its thickness or scale. Rounded table values can make two nearby outputs look equal when they are not. Pay attention to the domain, the direction of approach, and whether the exact function value is relevant to the question.
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.