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2.Algebraic Limits
Class 11 • Mathematics • NCERT
Algebraic Limits
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This lesson covers algebraic properties of limits as prescribed in NCERT Class 11. Students learn how to evaluate limits using standard limit laws for sums, products, and quotients of functions.
Lesson Objectives
- Understand algebraic properties of limits.
- Evaluate limits using limit laws.
- Solve limits of polynomials and rational functions.
- Build confidence for derivative concepts.
1. Algebraic Properties of Limits
If limx→a f(x) and limx→a g(x) exist, then limits of their algebraic combinations also exist.
These properties help evaluate limits easily.
2. Limit of a Sum or Difference
The limit of a sum (or difference) is the sum (or difference) of the limits.
lim (f(x) ± g(x)) = lim f(x) ± lim g(x)
3. Limit of a Product
The limit of a product of two functions is the product of their limits.
lim [f(x)g(x)] = (lim f(x))(lim g(x))
4. Limit of a Quotient
The limit of a quotient exists provided the limit of the denominator is not zero.
lim [f(x)/g(x)] = (lim f(x))/(lim g(x)), if lim g(x) ≠ 0
5. Limits of Polynomial Functions
For polynomial functions, the limit at x = a is obtained by direct substitution.
limx→2 (x² + 3x + 1) = 4 + 6 + 1 = 11
6. Limits of Rational Functions
If the denominator is not zero at x = a, substitute directly.
limx→1 (x² + 1)/(x + 2) = 2/3
Practice Questions (NCERT)
- State the algebraic property for sum of limits.
- State the algebraic property for product of limits.
- When does the limit of a quotient exist?
- Evaluate limx→2 (x + 5).
- Evaluate limx→1 (3x² − x).
- Evaluate limx→0 (2x + 7).
- Evaluate limx→1 (x² + 2x + 1).
- Evaluate limx→2 (x² − 4)/(x − 2).
- Can direct substitution always be used?
- Is this topic part of NCERT Class 11 syllabus?
✅ Show Answer Key
- lim(f+g) = lim f + lim g
- lim(fg) = (lim f)(lim g)
- When denominator limit ≠ 0
- 7
- 2
- 7
- 4
- Does not exist by substitution
- No
- Yes
© Aviate Learning – Limits & Derivatives (NCERT Class 11)