A. Rate of Change · Limits and Rates of Change

Limits and Rates of Change: Unit Review

A1.1 A1.2 A1.3 A1.4 A1.5 A1.6About 40 minutes

Learning goals

I can describe and calculate average and instantaneous rates of change, and connect them to secants and tangents.
I can find limits of sequences and functions numerically, graphically and algebraically, including indeterminate forms and limits involving infinity.
I can find derivatives at a point from first principles, compare methods, and decide where a function is differentiable.

Words to know

limit
The value a function or sequence approaches; written lim (x → a) f(x) = L or lim (n → ∞) tₙ = L.

difference quotient
(f(a + h) − f(a))/h: the average rate of change over [a, a + h].

derivative at a point
f′(a) = lim (h → 0) (f(a + h) − f(a))/h: the instantaneous rate of change and tangent slope at x = a.

Let's review!

In this unit you described rates of change in many forms, connected average rates to secants and instantaneous rates to tangents, and estimated instantaneous rates with shrinking intervals. Then you built the idea that makes calculus work, the limit: for sequences (the golden ratio, e), for functions (graphs, tables, one-sided limits, asymptotes), with limit laws and continuity, and algebraically for the indeterminate form 0/0. Finally you put it all together: the difference quotient as a limit, two methods compared, the derivative f′(a) from first principles, and where it fails to exist.

The next unit, Derivatives and Differentiation Rules, turns f′(a) into a function and finds shortcuts. This review has questions from every lesson. Try each one, then press Show answer to check.

How to use this review

Work through two or three sections at a time. For every limit, substitute first and decide what you got: a number, (non-zero)/0, or 0/0. For every rate, give units and say whether it is average or instantaneous. Check algebraic answers with a calculator at x = a ± 0.001 (radian mode for trigonometric functions). For each question you miss, write down why, and revisit that lesson. Finish with the 10-question quiz.

How the unit fits together: rates of change, then limits, then the derivative at a point.

The whole unit in one picture: secants from P to points Q that slide toward P, and the tangent they approach. Its slope is f′(a).

Just for fun

You started this unit asking how fast things change and ended it with an idea that took mathematicians well over a century to pin down: Newton and Leibniz used the idea informally in the late 1600s, but precise definitions of limits only arrived in the 1800s. You learned it in one unit. Not bad.

Rates of Change in the Real World

A town's population went from 18 200 in 2019 to 19 700 in 2025. Find the average rate of change.

Average or instantaneous? "At noon the lake was warming at 0.4 °C/h."

Find the average velocity of h(t) = 45 − 4.9t² over [1, 2].

Secants, Tangents and Smooth Functions

Find the secant slope of f(x) = x³ over [1, 2].

Why is there no instantaneous rate of change at the tip of a V-shaped graph?

What does "smooth" mean in this course?

Estimating Instantaneous Rates from Average Rates

Estimate the instantaneous rate of change of f(x) = x³ at x = 1 with a centred interval and h = 0.1.

A table gives d(2) = 11.0 m and d(4) = 19.4 m. Estimate the velocity at t = 3 s.

For a curve bending upward, which secant is steeper than the tangent: preceding or following?

Limits of Sequences: Fibonacci, the Golden Ratio and e

Find lim (n → ∞) (5n² − 3)/(2n² + n).

What do the ratios of consecutive Fibonacci numbers approach?

Does 1, −1, 1, −1, … converge?

The Limit of a Function: Graphs and Tables

f(x) = 3 − x for x < 1 and f(x) = x² + 1 for x ≥ 1. Find lim (x → 1) f(x).

A graph has a hole at (4, 7) and a filled dot at (4, 2). Find lim (x → 4) f(x) and f(4).

Estimate lim (x → 0) (sin 2x)/x with x = ±0.01 (radians).

Limits Involving Infinity and Asymptotes

Find lim (x → ∞) (7x² − 1)/(3 − 2x²).

Find lim (x → 1⁺) 2/(x − 1) and lim (x → 1⁻) 2/(x − 1).

A model is P(t) = 300 + 200(0.5)ᵗ. Find the long-term value.

Limit Laws and Continuity

Evaluate lim (x → 2) (x² + 1)/(x + 3).

f(x) = (x² − 16)/(x − 4). Classify the discontinuity at x = 4.

Find k so f(x) = kx² for x < 2 and f(x) = x + 6 for x ≥ 2 is continuous.

Indeterminate Forms: Factoring and Simplifying

Evaluate lim (x → −4) (x² + 2x − 8)/(x + 4).

Evaluate lim (x → 2) (x³ − 8)/(x − 2).

Evaluate lim (x → 0) [1/(x + 5) − 1/5]/x.

Indeterminate Forms: Rationalizing and Other Strategies

Evaluate lim (x → 16) (√x − 4)/(x − 16).

Evaluate lim (x → 0) (√(x + 1) − 1)/x.

Does lim (x → 4) |x − 4|/(x − 4) exist?

The Difference Quotient as a Limit

What does lim (h → 0) ((5 + h)² − 25)/h represent? Evaluate it.

Identify f and a: lim (h → 0) (√(4 + h) − 2)/h.

For s(t) = 4.9t², find the velocity at t = 1 s.

Instantaneous Rates With and Without Simplifying

Evaluate ((2 + h)² − 4)/h for h = 0.01 without simplifying.

Simplify ((2 + h)² − 4)/h and give the exact rate at x = 2.

What is the instantaneous rate of change of x³ at x = a?

The Derivative at a Point: First Principles

Find f′(1) for f(x) = 4x² − 3x from first principles.

Find f′(2) for f(x) = 1/(x + 2).

Write the tangent to f(x) = 4x² − 3x at x = 1.

C′(50) = 18 for a cost function in dollars. Interpret.

When Does a Function Have No Tangent? Differentiability

Is f(x) = |2x − 6| differentiable at x = 3?

f(x) = x² for x ≤ 3 and f(x) = 6x − 9 for x > 3. Is f differentiable at 3?

Give a function that is continuous but not differentiable at 0.

Check your understanding

Quiz

Which is an average rate of change?

The instantaneous rate of change of a smooth function at x = a equals:

What is lim (n → ∞) (3n + 1)/(6n − 5)?

If lim (x → 2⁻) f(x) = 5 and lim (x → 2⁺) f(x) = 3, then lim (x → 2) f(x):

What is lim (x → ∞) (4x³ + x)/(2x³ − 7)?

Evaluate lim (x → 3) (x² − 9)/(x − 3).

Evaluate lim (x → 9) (√x − 3)/(x − 9).

lim (h → 0) ((2 + h)³ − 8)/h represents:

What is f′(3) for f(x) = x² − 4x?

10.

At which point is f(x) = |x + 1| not differentiable?

For parents and teachers

Use the review to find what to revisit:

Units missing, signs wrong, or average/instantaneous mixed up → Lesson 1 (Rates of Change in the Real World).
Unsure why there is no rate at a corner, or what smooth means → Lesson 2 (Secants, Tangents and Smooth Functions).
Dividing a centred interval by h instead of 2h, or using intervals that miss the point → Lesson 3 (Estimating Instantaneous Rates from Average Rates).
Wrong power of n, or confusion about sequences that oscillate → Lesson 4 (Limits of Sequences).
Giving f(a) instead of the limit, or checking only one side → Lesson 5 (The Limit of a Function: Graphs and Tables).
Sign errors next to vertical asymptotes, or arithmetic with ∞ → Lesson 6 (Limits Involving Infinity and Asymptotes).
Holes versus asymptotes, or piecewise continuity → Lesson 7 (Limit Laws and Continuity).
Factoring slips, or writing 0/0 = 0 → Lesson 8 (Indeterminate Forms: Factoring and Simplifying).
Conjugate errors, absolute values, cube roots → Lesson 9 (Indeterminate Forms: Rationalizing and Other Strategies).
Not recognizing which function and point a limit describes → Lesson 10 (The Difference Quotient as a Limit).
Expanding (a + h)³ wrongly, or trusting tiny-h calculator values → Lesson 11 (Instantaneous Rates With and Without Simplifying).
First-principles algebra or tangent equations → Lesson 12 (The Derivative at a Point: First Principles).
Thinking continuous means differentiable, or using the wrong piece on each side → Lesson 13 (Differentiability).

Looking ahead: the next unit, Derivatives and Differentiation Rules, finds f′(x) as a function and builds the power, sum, product and chain rules from the limit definition you now know.