Playbook
Limits
86 q - 2.04/paper - 55% HARD. The hardest chapter in the subject by rate, and unusually it does NOT cherry-pick: the limit pages and the continuity pages sit at the same difficulty. Four limit toolkits (existence and infinity, algebraic, trigonometric, exponential-logarithmic) feed three continuity pages (a single point, piecewise junctions, the [x] and |x| discontinuities).
- Questions in the bank
- 86
- q/paper (2024–25 shifts)
- 2.04
- Tagged HARD
- 55%
- Subtopics
- 7
Strand: Long Tail
When you’ll see it
A limit that comes out 0/0 or infinity/infinity, or a piecewise function carrying an unknown constant that is asked to be continuous.
How this chapter is tested
86 q, 2.04/paper, 55% HARD — the hardest chapter in the subject by rate. It is also the one chapter in the tail with nowhere to hide: the four limit pages run 44-67% HARD and the three continuity pages 50-58%, so every page is above the paper's overall 38.2% HARD line. There is no cheap corner to cherry-pick.
The two halves ask for different work. Continuity is really equation-solving wearing a calculus costume: write the left-hand limit, the right-hand limit and the value at the point, set all three equal, and solve for the one or two unknowns. It is the more mechanical of the two despite carrying a comparable HARD rate, and it is where a student with a reliable method banks the chapter's marks — 45 of the 90 questions are continuity problems.
Limit Evaluation is recognition, not computation. Almost every question is one of a short list of standard forms in disguise, and the win is deciding within about fifteen seconds which tool applies — factorise, rationalise, divide by the highest power, or quote a standard limit. At 1.8 minutes a question, a limit you have to experiment on has already cost you a question elsewhere.
Practical consequence: give this chapter a hard time cap. Two questions a paper at 56% HARD is four marks that will not come cheaply, and with no negative marking an unresolved limit is still worth a marked option rather than a blank.
The sub-skills
The distinct skills inside the chapter, in the order to learn them.
Limits — Existence, One-Sided Limits and Limits at Infinity
Compute the two sides separately whenever |x| or [x] is in play — x/(|x| + x^2) has no limit at 0 while |x|/(|x| + x^2) tends to 1 — and settle a ratio at infinity by its leading powers, replacing any finite sum by its closed form first.
Algebraic Limits — Factorisation, Rationalisation and the xⁿ − aⁿ Form
Substitute first; on 0/0 factor and cancel, rationalise (both floors if both carry surds, twice for a nested root), or quote (x^n - a^n)/(x - a) = n a^(n-1). Read [f(x) - f(a)]/(x - a) as f'(a), and remember a finite limit forces the numerator to vanish.
Trigonometric Limits — sin x/x and the 1 − cos x Family
sin x / x and tan x / x tending to 1, (1 - cos kx)/x^2 tending to k^2/2, an identity applied before the limit, the shift x = pi/2 - h, degrees converted by pi/180, and one more term of the series when the first order cancels.
Exponential, Logarithmic and 1^∞ Limits
(a^x - 1)/x tending to log a and log(1 + x)/x tending to 1; (bc)^x - b^x - c^x + 1 factors as (b^x - 1)(c^x - 1); t = a^x turns mixed exponents into algebra; and any 1^infinity form is e to the limit of (f - 1)g.
Continuity at a Point — Finding f(c) and the Parameter
Every 'find k' or 'find f(0)' stem is a limit from the pages above set equal to a value. Carry a parameter inside the formula through the standard limits as a symbol, and differentiate an integral with a variable upper limit by the chain rule.
Continuity of Piecewise Functions — Junction Conditions and Parameter Systems
Count the junctions first — two unknowns need two junctions — then write left = right = value at each, using the piece whose inequality owns the point for the value. Solve the inequality that defines the pieces before writing any equation.
Discontinuities of [x], |x| and sgn x — Counting the Points
[x] jumps at every integer and [g(x)] wherever g crosses one; (x - a)/|x - a| is a jump of size 2 that no f(a) can bridge; and a factor that vanishes at the jump swallows it, so [x] sin(pi x) is continuous everywhere.
Traps to expect
Distractor shapes this chapter reuses. The Traps page covers the patterns that cut across chapters.
Value instead of limit
The distractor is f(a) computed by direct substitution, which is only the answer when f is continuous there. If substitution gives an indeterminate form, the value and the limit are different numbers.
Comparing the wrong terms at infinity
For a ratio of polynomials as x tends to infinity, only the leading powers matter. The wrong option comes from comparing constant terms or from keeping a lower-order term that vanishes.
Modulus at zero
For expressions like |x| / x the one-sided limits are +1 and -1, so the limit does NOT exist. Both 1 and -1 will be offered; 'does not exist' is the answer.
One raised to infinity treated as one
1^infinity is an indeterminate form, not 1. It needs the exponential-limit treatment, and the option reading 1 is planted for exactly this slip.
Learn it before you drill it
This chapter has full teaching notes — foundations, worked examples, self-checks and a per-subtopic mastery checkpoint. Read the notes once, then drill subtopic by subtopic below.
Limits notesDrill every limits question
86 questions from the bank, scoped to 7 bundled subtopics.
Drill one subtopic at a time
The 7 subtopics this playbook covers, in catalog order.
- Limits — Existence, One-Sided Limits and Limits at InfinityDrill
- Algebraic Limits — Factorisation, Rationalisation and the xⁿ − aⁿ FormDrill
- Trigonometric Limits — sin x/x and the 1 − cos x FamilyDrill
- Exponential, Logarithmic and 1^∞ LimitsDrill
- Continuity at a Point — Finding f(c) and the ParameterDrill
- Continuity of Piecewise Functions — Junction Conditions and Parameter SystemsDrill
- Discontinuities of [x], |x| and sgn x — Counting the PointsDrill
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