What does it mean for a function to approach a value near a point?
Quick answer
A function’s limit near an input is the output value its nearby inputs get closer to. It describes what happens around the chosen input, not necessarily what the function outputs exactly there. For example, if nearby inputs make the outputs approach 4, the limit is 4—even if the function is undefined at that input or has a different value there. A table or graph can help you see the pattern. [S1]
The question, the details, the next step.
I keep mixing up a limit with plugging a number into a function. What does it mean for a function to approach a value near a point?
A limit asks: as the input gets closer and closer to a chosen number, what value do the outputs get closer to? The input’s target is the point you are approaching; the limit is the output value you are approaching. You look at function behavior near that input, rather than only evaluating the function at the input itself. A graph or a table of nearby inputs and outputs can help reveal the pattern. [S1]
For a first picture, imagine that inputs just below 2 produce outputs near 4, and inputs just above 2 do too. That suggests the function approaches 4 as the input approaches 2. The point itself is a separate question.
Can you show me a worked example where the limit and the value at the point are different?
Consider a function defined by one rule except at 2, where it is assigned a special value:
f(x) = (x² − 4)/(x − 2) when x ≠ 2; f(2) = 7.
For inputs other than 2, factor the numerator: x² − 4 = (x − 2)(x + 2). So for those nearby inputs, the formula simplifies to f(x) = x + 2. This simplification is valid for x ≠ 2; it does not replace the separately specified value f(2) = 7.
Now look at inputs close to 2:
- At
x = 1.9, the output is3.9. - At
x = 1.99, the output is3.99. - At
x = 2.01, the output is4.01.
The outputs get close to 4 as the inputs get close to 2. So the limit is 4, while the function’s value at 2 is 7. A limit can exist even when the function’s value at that point is different, or when the function is not defined there. [S1]
How do I work out what to write when a question gives me a graph or a table instead of a formula?
Keep the input and output roles separate. First identify the input value the question says to approach. Then inspect outputs for inputs on either side that get progressively closer to it. Ask what number those outputs appear to get closer to. A graph can show the nearby curve trend; a table can show it through listed values. Both can suggest a limit, but they show nearby behavior rather than automatically telling you the function’s exact value at the target input. [S1]
For example, if a table lists inputs 1.9, 1.99, 2.01, and 2.1, with outputs 3.9, 3.99, 4.01, and 4.1, the outputs point toward 4 as the inputs close in on 2. Then check the table or graph separately for the function’s value exactly at 2. It might be marked with a filled point at another height, or there might be no value shown there. The evidence in a rough graph or a short table suggests a pattern; it does not by itself establish every detail of the function.
What’s a simple way to check my answer and avoid saying the function equals the limit?
Write two statements, not one: “As x approaches [input], f(x) approaches [limit].” Then, separately, “At x = [input], f(x) = [point value]” if that value is given or defined. If the function has no value there, say that instead. For the example, write: “As x approaches 2, f(x) approaches 4; f(2) = 7`.”
As a check, ask yourself what each number refers to: the input value is what x approaches, and the limit is what the outputs approach. A graph or table is useful for spotting the behavior, but a visual estimate may not give an exact value. When a formula is available, use its algebra carefully and keep any restrictions on the input in view. The central distinction is that a limit describes nearby function behavior, while f(a) describes the output at the particular input a. [S1]
A practical checklist
Identify the input value the question says to approach.
Check outputs for inputs close to that value, using a graph, table, or formula.
Look for the output value those nearby outputs approach.
Check separately whether the function has a value at the target input.
State the limit and the function’s value as separate facts.
A few more useful answers
Can a limit exist if the function is not defined at the point?
Yes. A limit concerns what outputs do for nearby inputs, so the function need not have a value at the target input. [S1]
Does a graph’s point at the target input determine the limit?
Not by itself. Look at how the nearby curve behaves; the value marked at the exact input can differ from the limit. [S1]
What should I do if a table’s outputs do not seem to settle near one value?
Describe what the listed values show without claiming a specific limit from that table alone. A short table may not establish the full behavior near the input.
Sources
References used for this guide.
- [S1] 2.2 The Limit of a Function — Calculus Volume 1, OpenStax openstax.org · Checked