Limits describe what a function is doing as the input gets close to a certain value, even if the function is not defined exactly there. They are one of the main ideas that make calculus possible because they let us study change, motion, and behavior near tricky points. Continuity builds on limits by asking whether a function has any breaks, jumps, or holes.
Infinite behavior adds another important case by showing what happens when function values grow without bound near certain inputs.
To analyze a limit, students often compare the left-hand and right-hand behavior of a graph or expression as approaches a value. A function is continuous at when exists, exists, and both are equal. If one-sided limits disagree, the limit does not exist, and if values shoot upward or downward without bound, the function may have a vertical asymptote.
These ideas connect graphs, tables, and algebraic rules into one framework for understanding functions.
Understanding Limits & Continuity
Limits are often found by direct substitution first. Put the approaching input into the expression. If this produces an ordinary number, the limit is usually that number because polynomials, roots within their domains, and many familiar functions behave smoothly.
Limit laws justify working in pieces. The limit of a sum is the sum of the limits. The limit of a product is the product of the limits.
A quotient can be handled the same way when the denominator limit is not zero. These rules save time, but they do not solve every problem.
A result such as zero divided by zero is not an answer. It is an indeterminate form, meaning the expression needs more analysis.
Factoring is especially useful when zero divided by zero appears in a rational expression. A shared factor may cancel for every nearby input, even though it cannot be used at the original troublesome input. This creates a hole rather than a jump or an asymptote.
Rationalizing helps with expressions containing square roots. Multiplying by a carefully chosen conjugate can remove a difference of roots and expose the useful factors.
Students should state any restriction that remains after simplifying. The simplified rule can describe nearby behavior, while the original function may still be undefined at one point.
L'Hopital's rule is another tool, but it has strict conditions. It applies to quotients that produce zero divided by zero or infinity divided by infinity after substitution. Differentiate the numerator and denominator separately, then evaluate the new quotient.
The rule may need to be used more than once. It does not apply automatically to forms such as zero times infinity or infinity minus infinity.
Those forms must first be rewritten as a quotient or simplified in another way. In a course, teachers often expect algebraic methods before L'Hopital's rule because factoring and rationalizing show more clearly why the limit has its value.
Limits at infinity describe long-term behavior rather than behavior near one input. For rational functions, compare the highest powers in the numerator and denominator. If the denominator has the larger power, values approach zero.
If both have the same highest power, values approach the ratio of their leading coefficients. If the numerator has the larger power, the function may grow without bound or follow a slanted polynomial pattern.
This helps identify horizontal or other end-behavior asymptotes. Such models appear when a quantity levels off, such as a machine approaching a maximum output, or when an average rate becomes stable as more data is collected.
Continuity matters because many calculus results require it. A continuous model can represent a quantity changing without sudden gaps, such as position during ordinary motion or temperature over time. Piecewise functions need special care at their joining inputs.
Evaluate the left rule, the right rule, and the assigned function value. A jump means the sides head toward different heights. A removable break means the nearby pattern agrees but one value is missing or misplaced.
An infinite break occurs near an asymptote. Graphs can suggest these features, but algebra confirms them. Tables can miss a hole or a steep asymptote when the chosen inputs are not close enough.
Key Facts
- means gets arbitrarily close to as gets close to
- and must be equal for to exist
- A function is continuous at if exists, exists, and
- If or , then has infinite behavior near
- A common vertical asymptote occurs when , where and
- For a removable discontinuity, simplifying first can reveal the limit, such as for , so
Vocabulary
- Limit
- The value a function approaches as the input gets close to a given number.
- One-sided limit
- A limit found by approaching a point only from the left or only from the right.
- Continuous
- A function is continuous at a point if there is no break there and the limit equals the function value.
- Discontinuity
- A point where a function has a break, hole, jump, or other failure of continuity.
- Vertical asymptote
- A vertical line that a graph approaches when function values increase or decrease without bound.
Common Mistakes to Avoid
- Assuming the function value and the limit must always be the same, which is wrong because a limit can exist even when is missing or different.
- Checking only one side of a point, which is wrong because a two-sided limit exists only if the left-hand and right-hand limits match.
- Canceling terms at a discontinuity without noting the restriction, which is wrong because simplification can hide a hole where the original function was undefined.
- Saying a limit does not exist whenever the graph goes to infinity, which is wrong because this usually means the function has infinite behavior rather than approaching a finite number.
Practice Questions
- 1 Find .
- 2 Evaluate .
- 3 A graph has and . Explain whether exists and what this says about continuity at .