12.3 Evaluation of Limits (Algebraic)

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Solved Examples

  1. Check form: At x=2x=2, Num =8−16+8=0= 8 - 16 + 8 = 0, Den =4−4=0= 4 - 4 = 0. Form is 0/00/0.
  2. Factorize: Numerator: x(x2−4x+4)=x(x−2)2x(x^2 - 4x + 4) = x(x-2)^2. Denominator: (x−2)(x+2)(x-2)(x+2).
  3. Limit: lim⁡x→2x(x−2)2(x−2)(x+2)\lim_{x \to 2} \frac{x(x-2)^2}{(x-2)(x+2)} Cancel one (x−2)(x-2) (since x≠2x \neq 2): lim⁡x→2x(x−2)x+2\lim_{x \to 2} \frac{x(x-2)}{x+2}
  4. Substitute: 2(2−2)2+2=04=0\frac{2(2-2)}{2+2} = \frac{0}{4} = 0

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