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Mathematics I · Week 4 — Limits and rates of change
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Mathematics I · Week 4 — Limits and rates of change
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9 min read
2026-07-19
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Week 4 — limits and rates of change
Calculus starts with a question about behaviour near a point. Keep three ideas distinct: the function’s value, values approached nearby, and the rate at which the output changes. A clean interpretation is worth more than a memorised differentiation rule.
Three separate questions
Value: what is
f(a)? Limit: what do nearby values approach? Rate: how fast is the output changing at or across a time/input interval? These can have different answers.Learning objectives
By the end of this week, you should be able to:
- Separate a function value, a limit, and continuity at the same input.
- Diagnose why a two-sided limit fails using one-sided behaviour.
- Distinguish average rate from instantaneous rate and report both with units.
- Read the derivative definition as a limiting process rather than direct division by zero.
- Explain why continuity alone does not guarantee a derivative.
Prerequisites
- Week 2 domain checks and removable-hole reasoning after cancelling a common factor.
- Fraction simplification and the coordinate meaning of slope.
- Basic function notation and input-output units from earlier weeks.
Core content
Limits are about approach
lim(x→a) f(x) = L means that values of f(x) can be made arbitrarily close to L by taking x sufficiently close to a (without necessarily taking x=a). A limit may exist even if the function is undefined at that point. Conversely, a defined value at a point does not force the nearby behaviour to approach it.When one-sided limits are relevant, compare the left and right approach. A two-sided limit exists only when they agree.
| Situation | Limit behaviour | Function value |
|---|---|---|
| Removable hole | nearby values may approach one number | may be undefined or assigned differently |
| Jump | left and right approach different values | can still be defined at the point |
| Continuous point | approaches the value | limit equals function value |
A three-column limit audit
For a point
a, keep these questions in separate columns rather than letting a graph’s appearance answer all of them at once.| Question | What you inspect | Example with f(x)=(x²-1)/(x-1) for x≠1 |
|---|---|---|
Is f(1) defined by the original rule? | the formula or piecewise definition at the point | no: the denominator is zero |
| What do nearby values approach? | simplify only away from the point, table, or graph | x+1 approaches 2 |
| Is the function continuous at 1? | value exists, limit exists, and they agree | no, because the function value is missing |
The simplified expression
x+1 is useful for the nearby behaviour, but it does not retroactively define the original fraction at x=1. A removable hole is therefore not a contradiction: the limit can be perfectly definite while the original function has no value there.One-sided limits are a disagreement test
For a piecewise rule, calculate the approach from each side before announcing a two-sided limit. If
p(x)=1 for x<0 and p(x)=3 for x≥0, then the left-hand limit at zero is 1 and the right-hand limit is 3. Because they disagree, lim(x→0)p(x) does not exist—even though p(0)=3 is defined.“Does not exist” is a conclusion about a particular limit, not a statement that the graph has no values anywhere. Name the reason: left/right disagreement, unbounded behaviour, or oscillation. That is much more informative than the abbreviation alone.
Average versus instantaneous rate
For a changing quantity
s(t), the average rate from t=a to t=b is[s(b) - s(a)] / [b - a].It measures the slope of a secant line across an interval. The instantaneous rate at
a is the limit of that average rate as the interval shrinks toward zero; geometrically, it is tangent slope when the derivative exists.The derivative is not “just a formula.” It records a relationship between units. If distance is kilometres and time is hours, a derivative has kilometres per hour. A derivative of zero means a horizontal tangent at that instant; it does not mean the original distance is zero or that the function is globally constant.
Original worked example — keep units attached
If
V(t) measures water volume in litres after t minutes, then [V(b)-V(a)]/[b-a] has units litres per minute. A positive rate means volume is increasing; a negative rate means it is decreasing. If V'(5)=0, the volume is momentarily not changing at minute 5. It does not say the tank is empty.Reading a change question
- Name the input and output quantities with units.
- Decide whether the question asks across an interval (average) or at one instant (derivative).
- State the allowed domain and any meaningful time interval.
- Report the rate with units and a sentence of interpretation.
- Check whether the sign matches the described direction of change.
The limit-to-derivative bridge
Average rate uses two inputs,
a and a+h. As h becomes very small, the secant slope can approach a limiting tangent slope. The derivative is that limiting rate when it exists. The wording “approaches” matters: plugging in h=0 immediately would create division by zero, so the limit describes nearby non-zero intervals.Use a graph with care. A tangent slope is local; it says what happens near one input. A graph can have a zero tangent slope at a turning point and still rise or fall elsewhere.
Derive one rate from the definition
Let
f(x)=x². At a general input x, the average rate from x to x+h is[f(x+h)-f(x)]/h = [(x+h)²-x²]/h.Expand before cancelling:
[x²+2xh+h²-x²]/h = [2xh+h²]/h. For a non-zero interval h, this simplifies to 2x+h. Only after the cancellation has removed the division by h do we take the limit as h→0, giving f'(x)=2x.Two details matter:
his not set equal to zero during the fraction step; it represents a non-zero nearby change.- The final derivative is a new rule for local slope. At
x=3, the tangent slope is6, while the function value is9. A rate and a value answer different questions.
This derivation is also a template: write the change quotient, expand or factor enough to expose a removable
h, simplify for h≠0, then take the limiting behaviour.When a function has no derivative
Continuity is useful, but it does not guarantee differentiability. The graph of
f(x)=|x| is continuous at zero, yet its left-hand slopes approach -1 and right-hand slopes approach 1. There is no single tangent slope there, so f'(0) does not exist.Keep the hierarchy straight:
- If a function is differentiable at a point, it is continuous there.
- A function can be continuous at a point and still have a corner, cusp, or other failure of differentiability.
- A derivative can fail because the function is discontinuous, because the one-sided slopes disagree, or because slopes become unbounded.
This is why a local graph check is valuable after a symbolic calculation: ask whether a unique tangent direction is plausible at the point you are interpreting.
Watch the interval shrink
The visualizer starts with a secant line through two points on
f(x)=x². Advance through the stages and watch the second point move toward the first. Your task is to explain what is changing in the numerator, what is changing in the denominator, and why the ratio can approach a finite tangent slope without ever evaluating a 0/0 fraction.Loading Visualizer...
Say the limit in a sentence
“The interval gets smaller” is incomplete. Say: “For non-zero intervals whose width approaches zero, the secant slope approaches the tangent slope.” That preserves the one condition that prevents division by zero.
Key formulas and translation sheet
| Idea | Compact form | Read it as |
|---|---|---|
| Limit | lim(x→a) f(x) = L | Nearby outputs approach L; this does not by itself state f(a). |
Continuity at a | lim(x→a) f(x) = f(a) | The value exists, the limit exists, and the two agree. |
| Average rate | [s(b)-s(a)]/[b-a] | Secant slope across an interval. |
| Derivative definition | f'(x) = lim(h→0) [f(x+h)-f(x)]/h | Tangent slope from non-zero intervals shrinking toward zero. |
| Differentiability rule | differentiable ⇒ continuous | The reverse implication can fail at a corner or cusp. |
Common pitfalls
A limit and a function value answer different questions
The limit studies nearby inputs, while
f(a) is the rule’s assigned value at exactly a. Check both before claiming continuity.A secant is not yet a tangent
Average rate uses two distinct inputs. Instantaneous rate is what the secant slopes approach as the interval narrows; do not call a finite interval’s slope a derivative.
Units belong to the rate
If an output is measured in litres and an input in minutes, a derivative is litres per minute. The sign tells the local direction of change, not the quantity’s size.
The difference quotient never evaluates at h = 0
During the algebraic fraction,
h is non-zero. Simplify for nearby non-zero values, then take the limit. Cancelling h does not make the original fraction defined at zero.Continuous does not automatically mean differentiable
|x| has no break at zero, yet its left and right slopes differ there. Look for a single local tangent direction, not merely an unbroken graph.Retrieval ladder
- Give an example where a limit can exist while the function is undefined at the point.
- State whether a question asks for an average rate or instantaneous rate, and identify the wording that tells you.
- Attach units to a derivative for a real quantity you choose.
- Explain why a derivative of zero is local information rather than a claim that the whole function is constant.
- In the secant-to-tangent visualizer, pause at the shrinking-gap stage. Predict whether the slope is above or below 2 at the marked point, then advance and check the displayed value.
Five-minute exit check
You are ready when you can separate value, limit, and rate in one sentence each; calculate or describe a secant slope across an interval; and interpret a derivative with units and sign.
Retrieval practice
Choose any quantity measured over time. State its units, write the units of its average rate, and explain what a negative instantaneous rate would mean in the situation. Then explain how a removable hole can have a limit even though the original function is undefined there.