Playbook
Limits
93 q - 2.08/paper - 56% HARD. The hardest chapter in the subject by rate, and unusually it does NOT cherry-pick: both its subtopics sit above 54% HARD. Continuity at a Point (47 q) is parameter-hunting; Limit Evaluation (46 q) is technique recognition.
- Questions in the bank
- 93
- q/paper (2024–25 shifts)
- 2.08
- Tagged HARD
- 56%
- Subtopics
- 2
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
93 q, 2.08/paper, 56% HARD — the hardest chapter in the subject by rate. It is also the one chapter in the tail with nowhere to hide: Continuity at a Point runs 57% HARD across 47 q and Limit Evaluation Techniques runs 54% across 46 q, so both halves are above the paper's overall 38.4% HARD line. There is no cheap corner to cherry-pick.
The two halves ask for different work. Continuity at a Point 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 the higher HARD rate, and it is where a student with a reliable method banks the chapter's marks.
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.
The standard limits, cold
sin x / x and tan x / x tending to 1, (a^x - 1)/x tending to ln a, (1 + x)^(1/x) tending to e, and (x^n - a^n)/(x - a) tending to n a^(n-1). Most Limit Evaluation questions are one of these after one algebraic step.
Indeterminate-form triage
Name the form first: 0/0, infinity/infinity, infinity minus infinity, 0 times infinity, or 1 raised to infinity. The form dictates the method, and misnaming it is what turns a 40-second question into a four-minute one.
Algebraic reduction
Factorise and cancel for 0/0 in polynomials; rationalise when a surd sits in numerator or denominator; divide numerator and denominator by the highest power of x for limits at infinity.
One-sided limits and the continuity test
Continuity at x = a needs three things to agree: the left-hand limit, the right-hand limit and f(a). Test all three separately — a function can have a limit at a point and still be discontinuous there.
Parameter hunting
Given a piecewise f with unknowns a and b, continuity gives one equation per junction point. Two unknowns need two junctions, or one junction plus a differentiability condition. Set up the equations before touching algebra.
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.
Drill every limits question
93 questions from the bank, scoped to 2 bundled subtopics.
Drill one subtopic at a time
The 2 subtopics this playbook covers, in catalog order.
Related playbooks
Often paired with this one — the technique, the trap or the taxonomy overlaps. Drill these next.