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Evaluate the numerator as x 0^-.

Mathematics

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Answer

−∞-\infty

Step-by-step solution

4 steps

To evaluate the limit lim⁡x→0−x−1x2(x+2)\lim_{x \to 0^-} \frac{x-1}{x^2(x+2)}, we will analyze the behavior of the numerator and the denominator as xx approaches 0 from the left side.

  1. Evaluate the numerator as x→0−x \to 0^-.

    As xx approaches 0 from the left, x−1x-1 approaches 0−10-1.

    lim⁡x→0−(x−1)=−1\lim_{x \to 0^-} (x-1) = -1
  2. Evaluate the denominator as x→0−x \to 0^-.

    The denominator is x2(x+2)x^2(x+2).
    As xx approaches 0 from the left, x2x^2 will be a very small positive number (e.g., (−0.001)2=0.000001(-0.001)^2 = 0.000001). So, x2→0+x^2 \to 0^+.
    As xx approaches 0 from the left, x+2x+2 approaches 0+20+2.

    lim⁡x→0−x2=0+\lim_{x \to 0^-} x^2 = 0^+ lim⁡x→0−(x+2)=2\lim_{x \to 0^-} (x+2) = 2

    Therefore, the denominator x2(x+2)x^2(x+2) approaches (0+)(2)(0^+)(2), which is 0+0^+.

  3. Combine the results for the numerator and denominator.

    The limit becomes a negative number divided by a very small positive number.

    lim⁡x→0−x−1x2(x+2)=−10+\lim_{x \to 0^-} \frac{x-1}{x^2(x+2)} = \frac{-1}{0^+}
  4. Determine the final value of the limit.

    When a negative number is divided by a very small positive number, the result tends to negative infinity.

    −10+=−∞\frac{-1}{0^+} = -\infty

    The final answer is −∞-\infty.

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