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Maharashtra State Board · HSC Commerce · Class 11 (Std XI)

Class 11 Maths and Statistics Part 1 Chapter 7: Limits

Short notes, key concepts, quick revision and practice MCQs with answers

The limit of a function describes the value f(x) approaches as x approaches a point. It is the foundation of calculus. The chapter defines limits, states the algebra of limits and evaluates limits by direct substitution, factorisation and rationalisation, and gives the standard limits of algebraic, exponential and logarithmic functions.

Key concepts: Limits

Meaning of limit
lim(x→a) f(x) = l means f(x) gets as close to l as we like when x is close to a (x ≠ a). The limit may exist even when f(a) is not defined.
Left and right hand limits
The limit exists only if the left hand limit lim(x→a⁻) f(x) and the right hand limit lim(x→a⁺) f(x) are both equal.
Algebra of limits
The limit of a sum, difference, product or quotient (denominator limit ≠ 0) is the sum, difference, product or quotient of the limits; lim k f(x) = k lim f(x).
Direct method
If f is a polynomial or a rational function defined at a, then lim(x→a) f(x) = f(a).
Factorisation method
If substitution gives 0/0, factorise numerator and denominator, cancel the common factor (x − a) and substitute again.
Rationalisation method
If square roots give 0/0, multiply numerator and denominator by the conjugate of the surd expression.
Standard algebraic limit
lim(x→a) (xⁿ − aⁿ)/(x − a) = n aⁿ⁻¹.
Exponential and logarithmic limits
lim(x→0) (aˣ − 1)/x = log a; lim(x→0) (eˣ − 1)/x = 1; lim(x→0) (1 + x)^(1/x) = e; lim(x→0) log(1 + x)/x = 1.

Quick revision: important points

Practice MCQs with answers

  1. lim(x→2) (x² + 3x) equals:

    1. 6
    2. 8
    3. 10
    4. 12
    Show answer

    Answer: C. 10
    Direct substitution: 4 + 6 = 10.

  2. lim(x→3) (x² − 9)/(x − 3) equals:

    1. 0
    2. 3
    3. 6
    4. 9
    Show answer

    Answer: C. 6
    (x − 3)(x + 3)/(x − 3) = x + 3 → 6.

  3. lim(x→1) (x⁵ − 1)/(x − 1) equals:

    1. 1
    2. 4
    3. 5
    4. 0
    Show answer

    Answer: C. 5
    n aⁿ⁻¹ = 5 × 1⁴ = 5.

  4. lim(x→0) (eˣ − 1)/x equals:

    1. 0
    2. 1
    3. e
    4. Not defined
    Show answer

    Answer: B. 1
    Standard limit.

  5. lim(x→0) (2ˣ − 1)/x equals:

    1. 2
    2. log 2
    3. 1/2
    4. 0
    Show answer

    Answer: B. log 2
    lim (aˣ − 1)/x = log a.

All Class 11 Maths and Statistics chapters

Class 11 Maths and Statistics notes: all chapters