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lim_( vec(x)3)(-x+3)/(x^(2)-2x-3)

lim⁡x⃗3−x+3x2−2x−3 \lim _{\vec{x} 3} \frac{-x+3}{x^{2}-2 x-3}

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Full solution

Q. lim⁡x⃗3−x+3x2−2x−3 \lim _{\vec{x} 3} \frac{-x+3}{x^{2}-2 x-3}
  1. Factor Denominator: First, let's factor the denominator x2−2x−3x^2 - 2x - 3. This factors into (x−3)(x+1)(x - 3)(x + 1).
  2. Rewrite Limit Expression: Now, we rewrite the limit expression with the factored denominator. lim⁡x→3−x+3(x−3)(x+1)\lim_{x\to 3}\frac{-x+3}{(x-3)(x+1)}
  3. Simplify Numerator: We notice that the numerator −x+3-x + 3 can be simplified to −(x−3)-(x - 3).\newlineThis allows us to cancel out the (x−3)(x - 3) term in the numerator and denominator.
  4. Simplify Limit Expression: After canceling, the limit expression simplifies to: lim⁡x→3−1x+1\lim_{x\to 3}\frac{-1}{x+1}
  5. Substitute x=3x = 3: Now we can directly substitute x=3x = 3 into the simplified expression.−13+1=−14\frac{-1}{3+1} = -\frac{1}{4}

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