Defining Limits and Using Limit Notation: Study Guide (Calculus AB 1.2)

Course scope. This topic is shared by Calculus AB and Calculus BC. Everything here is examinable for both.

A note on notation

This page has no equation renderer, so limits are written in a compact form:

lim (x → 3) f(x) means “the limit as x approaches 3 of f(x)”.

On paper, write “lim” with “x → 3” underneath it, then f(x) to the right. Both forms mean the same thing.

What a limit describes

In Topic 1.1 you saw that an average rate of change needs two different inputs. At a single instant the change in the input is zero, so the average rate is undefined. Calculus gets round this by asking a different question: what value do the outputs head towards as the input gets close to a point? The answer is a limit.

Here is the idea in plain words.

Informal definition. Let f be a function and c a number. The limit of f(x) as x approaches c is the real number R if you can make f(x) as close to R as you like by choosing x close enough to c, but not equal to c.

If such a number R exists, you write

lim (x → c) f(x) = R

Three parts of the definition do most of the work.

1. “x close enough to c, but not equal to c.” You look at inputs on both sides of c, near c. You never use x = c itself.
2. “As close to R as you like.” It is not enough for f(x) to get near R. For any closeness you choose, however small, every input close enough to c (other than c itself) must give an f(x) at least that close to R.
3. “A real number R.” The limit is one number. If the outputs do not settle on one real number, the limit does not exist.

Reading and writing limit notation

Each symbol in lim (x → c) f(x) = R has a job.

Part	Read it as	What it tells you
lim	“the limit”	You are describing where outputs are heading, not one output
x → c	“as x approaches c”	The input value you are moving towards
f(x)	“of f of x”	The function whose outputs you watch
= R	“is R”	The single real number the outputs approach

So lim (x → −1) g(x) = 4 is read “the limit as x approaches −1 of g of x is 4”. In words: as x gets closer and closer to −1, the values of g(x) get as close as you like to 4.

An equivalent arrow form. You may also see “g(x) → 4 as x → −1”. It means exactly the same thing. Use whichever form the question uses.

The letter does not matter. lim (t → 2) f(t) and lim (x → 2) f(x) are the same number. The letter under “lim” must match the letter inside the function.

Notation you will meet next. Topic 1.3 adds one-sided limits, written with a small sign after c (x → c⁻ from the left, x → c⁺ from the right), and the shorthand “= ∞” for unbounded behaviour. A limit written as “= ∞” is still a limit that does not exist as a real number.

The limit ignores f(c)

The definition says “not equal to c”. This has a big consequence: the value f(c) plays no part in the limit. Look at this function.

g(x) = ½x² + 1 for x ≠ 2, and g(2) = 5.

The outputs near x = 2 crowd towards 3. The single point at height 5 does not change where they are heading. So

lim (x → 2) g(x) = 3, but g(2) = 5.

The same limit would be 3 if g(2) were 3, or if g(2) were not defined at all. Three cases can happen at x = c:

Case	Example	Limit at c	f(c)
Value matches limit	g at x = 0	1	1
Value differs from limit	g at x = 2	3	5
No value at all	½x² + 1 with x = 2 removed from the domain	3	undefined

One limit, three representations

A limit statement can be shown graphically, numerically or analytically. Here is the same fact, lim (x → 2) g(x) = 3, in each form.

Graphically. Figure 1: trace the curve towards x = 2 from the left and from the right. Both paths head to height 3.

Numerically. A table with inputs closing in on 2 from both sides:

x	1.9	1.99	1.999	2.001	2.01	2.1
g(x)	2.805	2.98005	2.9980005	3.0020005	3.02005	3.205

The table does not include x = 2, and it does not need to. The outputs close in on 3 from below and from above. A table suggests a limit. It cannot prove one, because it only shows finitely many inputs.

Analytically. In symbols, lim (x → 2) g(x) = 3. Here you can confirm it from the formula: for x ≠ 2, g(x) = ½x² + 1, and as x → 2 this heads to ½(4) + 1 = 3.

Good answers move between the forms. If you are given notation, you should be able to say what the graph must look like near c, and which table values would fit.

What “as close as you like” means (background)

This section is background. The formal epsilon-delta definition is not assessed on either exam. It is here only to make the words of the definition concrete.

Take f(x) = 3x + 1 near c = 2. The outputs approach 7. Suppose someone demands that f(x) be within 0.3 of 7. That means 6.7 < f(x) < 7.3. Solving gives 1.9 < x < 2.1. So every x within 0.1 of 2 does the job.

Demand more: within 0.003 of 7. Now you need x within 0.001 of 2, that is 1.999 < x < 2.001. Whatever closeness is demanded, a small enough window around 2 meets it. That is what “as close as you like” means.

Worked example 1: translating between words, notation and context

Question. A delivery drone climbs and then hovers. H(t) is its height in metres, t seconds after take-off.

(a) Write in limit notation: “As the time gets closer and closer to 8 seconds, the drone’s height gets as close as we like to 30 metres.” (b) Explain in words, with units, what lim (t → 12) H(t) = 30 means. (c) Does lim (t → 8) H(t) = 30 tell you the value of H(8)?

1. Find c and R in (a). The input is time t, and it approaches 8, so c = 8. The output is height, and it approaches 30, so R = 30.
2. Write the statement. lim (t → 8) H(t) = 30. Use t, the letter the question uses, both under “lim” and inside H.
3. Read the notation in (b). The input t approaches 12. The outputs H(t) approach 30. Add units and context: as the time gets closer to 12 seconds after take-off, from either side, the drone’s height gets as close as we like to 30 metres.
4. Apply the definition in (c). The limit uses times near 8, never t = 8 itself. So the statement alone does not give H(8).

Answers. (a) lim (t → 8) H(t) = 30. (b) As t approaches 12 s, the height approaches 30 m. (c) No.

Interpretation. A real drone’s height changes smoothly, so in practice H(8) will also be 30 m. But that conclusion uses extra information about the drone (that its height has no sudden jumps). It does not come from the limit statement. Topic 1.11 makes this idea precise as continuity.

Worked example 2: deciding which limit statements are true

Question. Use g from Figure 1: g(x) = ½x² + 1 for x ≠ 2, and g(2) = 5. Decide whether each statement is true or false, with a reason.

(i) lim (x → 2) g(x) = 5 (ii) lim (x → 2) g(x) = 3 (iii) g(2) = 3 (iv) lim (x → 0) g(x) = g(0) (v) lim (x → 3) g(x) = 5.5

1. (i) False. For x near 2 but not equal to 2, g(x) = ½x² + 1 is close to 3, not 5. The value 5 is only g(2), which the limit ignores.
2. (ii) True. The graph heads to height 3 from both sides, and the table values close in on 3.
3. (iii) False. g(2) is defined separately as 5. The statement mixes up the value at 2 with the limit at 2.
4. (iv) True. Near 0 the outputs approach ½(0)² + 1 = 1, and g(0) = 1. Here the limit and the value happen to match.
5. (v) True. Near 3 the formula ½x² + 1 applies on both sides, and the outputs approach ½(9) + 1 = 5.5.

Check. Compare (ii) and (iii). The same two numbers, 2 and 3, appear in both. One statement is about where outputs head; the other is about a single output. Only the first is true.

Common misconceptions

• “The limit is the value at c.” The limit uses only x near c, with x ≠ c. The value f(c) may differ or may not exist.
• “If f(c) is undefined, the limit does not exist.” Not so. A missing point (a hole) does not stop nearby outputs from approaching a number.
• Leaving out “x → c”. “lim f(x) = 3” does not say where x is heading. The limit of the same function can be different at different points.
• Dropping “lim”. “(x → 2) g(x) = 3” or “g(x) = 3” is a different (and false) statement. Keep “lim” on every line.
• Swapping c and R. In lim (x → 2) g(x) = 3, the 2 is an input and the 3 is an output. Writing lim (x → 3) g(x) = 2 means something else entirely.
• “Getting near R is enough.” The outputs must get as close as you like to R, not just within some fixed distance. A number slightly off, such as 3.01, is not the limit here.
• “A table proves the limit.” A table suggests a value. Use the formula, a graph or later results (Topics 1.5 and 1.6) to justify it.
• Mismatched letters. lim (x → 2) f(t) is not well formed. The variable under “lim” must be the input of the function.

Where this leads

You now have the language. Topic 1.3, Estimating Limit Values from Graphs, uses it to read limits from graphs, including one-sided limits and limits that do not exist. Topic 1.4 does the same with tables. If you need a reminder of why limits are needed at all, revisit Topic 1.1, Introducing Calculus: Can Change Occur at an Instant?. Students in either course can return to the Calculus AB roadmap or the Calculus BC roadmap.

Try the practice questions now, then use the revision notes and the checklist to consolidate.