Limits by factorisation: Class 11 practice
If direct substitution gives 0/0, factorise and cancel the common factor (x − a), then substitute. Read the method, follow the three worked examples, then try the six questions.
What you should know first
- Factorising quadratics
The idea behind limits by factorisation
If direct substitution gives 0/0, factorise and cancel the common factor (x − a), then substitute. The standard result lim (xⁿ − aⁿ)/(x − a) = naⁿ⁻¹ is a shortcut.
Step-by-step method
- Substitute x = a; if you get 0/0, factorise.
- Cancel (x − a).
- Substitute again.
Worked example 1 (Basic)
Find lim x→-5 (x² + 7x + 10)/(x + 5).
- Factor: (x + 5)(x + 2)
- Cancel, then x = -5: -5 + 2 = -3
- Answer: -3
Worked example 2 (Standard)
Find lim x→5 (x² − 7x + 10)/(x − 5).
- Factor: (x − 5)(x − 2)
- Cancel, then x = 5: 5 − 2 = 3
- Answer: 3
Worked example 3 (Challenge)
Find lim x→2 (x^4 − 16)/(x − 2).
- Standard limit: n a^(n − 1)
- = 4 × 2^3 = 32
- Answer: 32
Common mistakes and misconceptions
- Writing 0/0 = 0 or 1
- Cancelling terms rather than factors
Practice questions
- (Basic) Find lim x→2 (x² − 6x + 8)/(x − 2).
- (Basic) Find lim x→-5 (x² + 4x − 5)/(x + 5).
- (Standard) Find lim x→1 (x² − 2x + 1)/(x − 1).
- (Standard) Find lim x→5 (x² − 3x − 10)/(x − 5).
- (Challenge) Find lim x→2 (x^6 − 64)/(x − 2).
- (Challenge) Find lim x→3 (x^4 − 81)/(x − 3).
Answer key with full working
- Q1: Factor: (x − 2)(x − 4) → Cancel, then x = 2: 2 − 4 = -2 → Answer: -2
- Q2: Factor: (x + 5)(x − 1) → Cancel, then x = -5: -5 − 1 = -6 → Answer: -6
- Q3: Factor: (x − 1)(x − 1) → Cancel, then x = 1: 1 − 1 = 0 → Answer: 0
- Q4: Factor: (x − 5)(x + 2) → Cancel, then x = 5: 5 + 2 = 7 → Answer: 7
- Q5: Standard limit: n a^(n − 1) → = 6 × 2^5 = 192 → Answer: 192
- Q6: Standard limit: n a^(n − 1) → = 4 × 3^3 = 108 → Answer: 108
Syllabus note
This skill is listed under "Limits and Derivatives" for Class 11 in the NCERT/CBSE books. ICSE, IGCSE, IB and state boards may teach it in a different chapter or year.
Answers on this page are generated and automatically re-checked by a second, independent calculation. They have not been reviewed by a teacher — if you spot a problem, tell us on WhatsApp.
Related practice and notes
Frequently asked questions
Which chapter is limits by factorisation in?
"Limits and Derivatives", Class 11 Maths (NCERT/CBSE). Other boards may place it differently.
How were the answers checked?
Answers on this page are generated and automatically re-checked by a second, independent calculation. They have not been reviewed by a teacher — if you spot a problem, tell us on WhatsApp.