Currently Empty: $0.00
3.Limits Using Standard Results
Class 11 • Mathematics • NCERT
Limits Using Standard Results
[ Embed Your Video Lecture Here ]
This lesson focuses on evaluating limits using standard results prescribed by NCERT Class 11. Students learn to apply these limits effectively by transforming expressions into standard forms.
Lesson Objectives
- Memorise NCERT standard limits.
- Apply limits by converting expressions to standard form.
- Evaluate trigonometric limits safely.
- Avoid common exam mistakes.
1. What Are Standard Limits?
Standard limits are limits whose values are known and frequently used to evaluate more complex limits.
NCERT requires students to remember these results.
2. Important Standard Results (NCERT)
limx→0 (sin x)/x = 1
limx→0 (tan x)/x = 1
limx→0 (1 − cos x)/x² = 1/2
limx→0 (ex − 1)/x = 1
limx→0 (ax − 1)/x = ln a
limx→0 (tan x)/x = 1
limx→0 (1 − cos x)/x² = 1/2
limx→0 (ex − 1)/x = 1
limx→0 (ax − 1)/x = ln a
3. Converting Expressions into Standard Form
Most problems require algebraic manipulation to match a standard result.
Example:
limx→0 (sin 3x)/(3x) × 3 = 3
limx→0 (sin 3x)/(3x) × 3 = 3
4. Trigonometric Limits
Trigonometric limits must be converted so that the angle tends to zero.
Angles must be in radians for standard limits.
5. Simple NCERT Examples
• limx→0 (sin 5x)/(x) = 5
• limx→0 (1 − cos 2x)/(x²) = 2
Practice Questions (NCERT)
- State limx→0 (sin x)/x.
- State limx→0 (1 − cos x)/x².
- Evaluate limx→0 (sin 7x)/(x).
- Evaluate limx→0 (tan 3x)/(x).
- Evaluate limx→0 (1 − cos 4x)/(x²).
- Why must angles be in radians?
- Evaluate limx→0 (ex − 1)/x.
- Evaluate limx→0 (ax − 1)/x.
- Are standard limits part of NCERT?
- Are these limits required for derivatives?
✅ Show Answer Key
- 1
- 1/2
- 7
- 3
- 8
- Because standard limits hold in radians
- 1
- ln a
- Yes
- Yes
© Aviate Learning – Limits & Derivatives (NCERT Class 11)