Limits and Rate of Change
FoundationA limit describes the value a function approaches as the variable approaches a number; with direct substitution, factorisation and rationalisation, limits at infinity by comparing degrees, two standard limits, and the move from average to instantaneous rate of change, you are ready for differentiation.
- A limit describes the value a function approaches as x approaches a; the function need not be defined at a.
- Try direct substitution first; for zero over zero, factorise or rationalise and cancel before substituting.
- For limits at infinity, compare highest degrees: equal degrees give the ratio of leading coefficients, a lower-degree numerator gives zero, a higher-degree numerator means no limit.
文體小知識：數學概念說明文——講解極限的直觀概念與四種求法（直接代入、因式分解、有理化、比較次數），再以兩個標準極限與平均變化率收結，是微積分的入門課。
§Text Passage
1Calculus is the mathematics of change. A car covers thirty kilometres on a motorway in half an hour — an average speed of sixty kilometres per hour — yet the speedometer reading changes from moment to moment. To describe how fast the car moves at one instant, we must compare the distance travelled over an ever-shorter interval of time; this idea is where limits begin, and it opens the whole story of calculus.
2A limit describes the value a function approaches as its variable approaches a number. For example, for the function whose value is x squared, as x approaches 3 the function values approach 9, so we say the limit of the function there is 9. Note that a limit describes the trend of approaching: whether the function is defined at that point, or equals the limit there, does not affect the existence of the limit.
3The first technique is direct substitution. The curriculum lists limit theorems for sums, differences, products, quotients, scalar multiples and composite functions, all usable without proof: the limit of each part is found and then combined. The quotient theorem requires the denominator's limit to be non-zero. Most algebraic functions are continuous, so if substitution gives a definite number, that number is the limit.
4If substitution makes both numerator and denominator approach zero, giving the indeterminate form zero over zero, we simplify first. For example, as x approaches 2, divide x squared minus four by x minus two: substituting gives zero over zero. Factorise the numerator into the product of x plus two and x minus two, cancel the common factor to leave x plus two, and substitution gives four. Limits involving surds are often handled by rationalisation: multiply numerator and denominator by the conjugate surd, clear the root, then substitute.
5As x tends to infinity, compare the highest degrees of numerator and denominator: equal degrees give the ratio of the leading coefficients, while a lower-degree numerator gives a limit of zero. Two standard limits are worth memorising: as the angle approaches zero, the sine of the angle divided by the angle tends to one; as x approaches zero, e to the power x minus one, divided by x, also tends to one. Taking the limit of the average rate of change gives the instantaneous rate — the limit is the cornerstone of the derivative in the next lesson.
1Summary 重點摘要
A limit describes the value a function approaches as the variable approaches a number; with direct substitution, factorisation and rationalisation, limits at infinity by comparing degrees, two standard limits, and the move from average to instantaneous rate of change, you are ready for differentiation.
2Vocabulary cards Vocabulary
The number a function approaches as its variable approaches a given value, written lim x→a f(x).
The process of a variable or function value getting arbitrarily close to a number without necessarily reaching it.
Finding a limit by substituting the value into the function; if the result is a definite number, that number is the limit.
A form such as zero over zero obtained by substitution, where the limit is not yet determined and simplification is needed.
Writing a polynomial as a product of factors so that common factors can be cancelled to remove zero over zero.
Multiplying numerator or denominator by a conjugate surd to remove the square root, using the difference of squares.
Theorems for limits of sums, differences, products, quotients, scalar multiples and composite functions, usable without proof.
The behaviour of a function as x grows without bound; for rational functions, judged by comparing the highest degrees of numerator and denominator.
lim θ→0 (sin θ/θ) = 1 and lim x→0 ((eˣ−1)/x) = 1, the keys to limits of trigonometric and exponential functions.
The change in function value divided by the change in the variable, [f(b)−f(a)]/(b−a), equal to the slope of the secant line.
3Step by step Steps
- Step 1: The intuitive idea of a limitLimits are about approaching, not arriving. Suppose the function value is x squared. As x approaches 3 through 2.9, 2.99, 2.999, the values are 8.41, 8.9401, 8.994 — creeping towards 9; approaching from the right through 3.1, 3.01 works the same way. For example, no matter whether x approaches 3 from the left or the right, the values approach 9 without bound, so we say the limit as x→3 is 9, written lim x→3 x² = 9. Even if the function is undefined at the point, or its value there differs from the limit, the trend of approaching still exists — the most basic principle of limits.
- Step 2: Limit theorems and direct substitutionThe curriculum lists six limit theorems — for sums, differences, products, quotients, scalar multiples and composite functions — all usable without proof. Limits can be found part by part: the limit of a product is the product of the limits, the limit of a quotient is the quotient of the limits, but for quotients the denominator's limit must be non-zero. For example, lim x→2 (3x²+1): substituting gives 3×4+1 = 13. A polynomial's limit at any point equals its value there, so the first move in almost every question is to try direct substitution.
- Step 3: Zero over zero and factorisationIf direct substitution gives zero over zero, the limit is not necessarily absent — it is a hint to simplify first. For example, for lim x→3 (x²−9)/(x−3), substitution gives zero over zero; factorise the numerator as (x+3)(x−3). Since x approaches 3 but x ≠ 3, we may cancel (x−3), leaving x+3, and substitution gives 6. Factorise, cancel, substitute — the standard three steps for zero over zero. Most limits of one polynomial divided by another yield to this technique; just remember to substitute only after cancelling.
- Step 4: Rationalising to remove surdsLimits involving surds are handled by rationalisation: multiply numerator and denominator by the conjugate surd, using (a−b)(a+b) = a²−b² to expand the root into a difference. For example, for lim x→0 (√(x+9)−3)/x, multiplying by the conjugate (√(x+9)+3) turns the numerator into (x+9)−9 = x, so the expression becomes 1/(√(x+9)+3), and substitution gives one sixth. The choice of conjugate follows a pattern: if the expression is a surd minus a constant, multiply by surd plus constant, and vice versa — the single goal is to turn the root into a difference of squares.
- Step 5: Limits at infinityWhen x tends to infinity, direct substitution is meaningless; compare the highest degrees of numerator and denominator instead. Equal degrees: the limit is the ratio of the leading coefficients. Numerator of lower degree: the limit is zero. Numerator of higher degree: the values grow without bound and the limit does not exist. For example, lim x→∞ (3x²−x+1)/(2x²+5) = 32 because both highest terms are quadratic, while (5x+1)/x² tends to zero. In practice, divide numerator and denominator by the highest power of x: some terms vanish to zero, the coefficients remain, and the answer is obvious.
- Step 6: The two standard limitsTwo standard limits unlock trigonometric and exponential limits and must be memorised: lim θ→0 (sin θ/θ) = 1 and lim x→0 ((eˣ−1)/x) = 1. The key is reshaping the expression into the standard form. For example, for lim x→0 (sin 5x)/x, write it as 5 × (sin 5x)/(5x); the second factor is exactly the standard form and tends to 1, so the limit is 5. Similarly (sin 3x)/(sin 5x) tends to 35. When reshaping, keep the coefficients consistent front and back — the most common trap in exams.
- Step 7: From average to instantaneous rate of changeThere are two kinds of rate of change. The average rate of change is [f(b)−f(a)]/(b−a), the slope of the secant joining two points on the curve. The instantaneous rate of change is the limit of the average rate as the interval shrinks to zero — the slope of the tangent at a point. For example, a car covering 120 kilometres in two hours has an average speed of 60 km/h, yet the speedometer's instantaneous reading changes constantly; to know the speed at one instant, let the time interval shrink towards zero and take the limit. That limit is precisely the derivative of the next lesson.
5Worked example Example
Example: evaluate the limit lim x→0 (√(x+9)−3)/x.
- Substituting x = 0 gives (3−3)/0, the indeterminate form zero over zero, so simplify first
- Multiply numerator and denominator by the conjugate surd (√(x+9)+3): the expression becomes [(√(x+9)−3)(√(x+9)+3)]/[x(√(x+9)+3)]
- By the difference of squares, the numerator is (x+9)−9 = x, so the expression is x/[x(√(x+9)+3)] = 1/(√(x+9)+3) for x ≠ 0
- Substitute x = 0: the limit is 1/(√9+3) = 16
- Answer: The limit is 16. For a zero-over-zero limit involving surds, rationalise the numerator first, cancel the factor causing zero, and only then substitute; keep the justification that x approaches 0 but x ≠ 0 at every step.
6Common mistakes Mistakes
7Quick check Quick check
Q1What does the limit lim x→a f(x) describe?
Q2Evaluate lim x→2 (x²−4)/(x−2).
Q3Evaluate lim x→∞ (5x²+3x)/(2x²+1).
Q4Evaluate lim x→0 (sin 4x)/x.
Q5Let f(x) = x². Find the average rate of change of f from x = 1 to x = 4.
8Recap Recap
- A limit describes the value a function approaches as x approaches a; the function need not be defined at a.
- Try direct substitution first; for zero over zero, factorise or rationalise and cancel before substituting.
- For limits at infinity, compare highest degrees: equal degrees give the ratio of leading coefficients, a lower-degree numerator gives zero, a higher-degree numerator means no limit.
- Memorise the standard limits lim θ→0 (sin θ/θ) = 1 and lim x→0 ((eˣ−1)/x) = 1, and reshape expressions into these forms.
- Average rate of change is the secant slope; shrinking the interval and taking the limit gives the instantaneous rate — the definition of the derivative in the next lesson.