Limits
Maharashtra Board · Class 11 · Mathematics & Statistics
Quick revision notes for Limits — Maharashtra Board Class 11 Mathematics & Statistics. Key concepts, formulas, and definitions for last-minute revision.
Key Topics to Revise
Definition and Basic Concepts of Limits
- A limit describes the value a function approaches as the input approaches a specific value
- The notation lim(x→a) f(x) = L means f(x) approaches L as x approaches a
- x → a means x takes values very close to 'a' but never equals 'a'
Algebra of Limits
- Limit of sum equals sum of limits: lim[f(x) + g(x)] = lim f(x) + lim g(x)
- Limit of product equals product of limits: lim[f(x) × g(x)] = lim f(x) × lim g(x)
- Limit of quotient equals quotient of limits (when denominator limit ≠ 0)
Direct Substitution Method
- Simplest method: substitute x = a directly in f(x)
- Works when f(x) is continuous at x = a
- Applicable for polynomial functions, rational functions where denominator ≠ 0
Factorization Method
- Used when direct substitution gives 0/0 form
- Factor both numerator and denominator to find common factors
- Cancel common factors (since x ≠ a, the factor (x-a) ≠ 0)
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