Limits involving sin x / x: Class 11 practice
For x in radians, lim x→0 (sin x)/x = 1. Read the method, follow the three worked examples, then try the six questions.
What you should know first
- Radian measure
The idea behind limits involving sin x / x
For x in radians, lim x→0 (sin x)/x = 1. Adjust the angle so it matches: (sin kx)/x = k × (sin kx)/(kx) → k.
Step-by-step method
- Rewrite so the angle inside sin matches the denominator.
- Use lim (sin u)/u = 1.
- Multiply by the constants you introduced.
Worked example 1 (Basic)
Find lim x→0 (sin 8x)/x.
- = 8 × (sin 8x)/(8x)
- → 8
- Answer: 8
Worked example 2 (Standard)
Find lim x→0 (sin 8x)/(sin 4x).
- = (8/4) × [(sin 8x)/(8x)] ÷ [(sin 4x)/(4x)]
- → 2
- Answer: 2
Worked example 3 (Challenge)
Find lim x→0 (tan 5x)/(8x).
- tan u/u → 1
- = 5/8
- Answer: 5/8
Common mistakes and misconceptions
- Using degrees
- Forgetting to balance the constant
Practice questions
- (Basic) Find lim x→0 (sin 2x)/x.
- (Basic) Find lim x→0 (sin 7x)/x.
- (Standard) Find lim x→0 (sin 2x)/(sin 9x).
- (Standard) Find lim x→0 (sin 6x)/(sin 3x).
- (Challenge) Find lim x→0 (tan 5x)/(3x).
- (Challenge) Find lim x→0 (tan 6x)/(4x).
Answer key with full working
- Q1: = 2 × (sin 2x)/(2x) → → 2 → Answer: 2
- Q2: = 7 × (sin 7x)/(7x) → → 7 → Answer: 7
- Q3: = (2/9) × [(sin 2x)/(2x)] ÷ [(sin 9x)/(9x)] → → 2/9 → Answer: 2/9
- Q4: = (6/3) × [(sin 6x)/(6x)] ÷ [(sin 3x)/(3x)] → → 2 → Answer: 2
- Q5: tan u/u → 1 → = 5/3 → Answer: 5/3
- Q6: tan u/u → 1 → = 3/2 → Answer: 3/2
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 involving sin x / x 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.