Example 1 · level 1
Evaluate lim (x → −2) (x³ + 2x² + 2x + 4).
Show solution
- Substituting x = −2 gives 0.
Answer: 0
CLASS 11 MATHS · LIMITS AND DERIVATIVES
Limits of polynomial and rational functions
For a polynomial, substitute directly. If substitution gives 0/0, factorise and cancel the common factor, or use lim (xⁿ − aⁿ)/(x − a) = naⁿ⁻¹.
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Worked examples
Example 2 · level 2
Evaluate lim (x → 5) (x² − 8x + 15)/(x − 5).
Show solution
- After cancelling, the limit is lim (x → 5) (x − 3).
- = 5 − 3 = 2.
Answer: 2
Example 3 · level 3
Evaluate lim (x → 3) (x² − 9)/(x⁴ − 81).
Show solution
- Limit = (2 × 3)/(4 × 27) = 6/108 = 1/18.
Answer: 1/18