Computing Limits with Algebraic Simplification

When direct substitution leads to an indeterminate form like 00\frac{0}{0}, simplify the expression algebraically.

Example:

limx→3x2−9x−3\lim_{x \to 3} \frac{x^2 - 9}{x - 3}

Steps:

  1. Factor the numerator:
(x−3)(x+3)x−3\frac{(x - 3)(x + 3)}{x - 3}
  1. Cancel (x−3)(x - 3):
x+3x + 3
  1. Plug in x=3x = 3:
3+3=63 + 3 = 6

The simplification resolves the indeterminate form and reveals the limit.

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