Determine the following limits.


b. lim x→−2^− (x − 4) / x(x + 2)

Verified step by step guidance

1. Step 1: Identify the type of limit. Since the limit is as x approaches -2 from the left (x \to -2^-), we need to consider the behavior of the function as x gets closer to -2 from values less than -2.
2. Step 2: Analyze the function \( \frac{x - 4}{x(x + 2)} \). Notice that the denominator becomes zero when x = -2, which suggests a potential vertical asymptote at x = -2.
3. Step 3: Determine the sign of the function as x approaches -2 from the left. For x slightly less than -2, both x and (x + 2) are negative, making the denominator positive. The numerator (x - 4) is negative, resulting in the entire fraction being negative.
4. Step 4: Consider the magnitude of the function as x approaches -2 from the left. As x gets closer to -2, the denominator approaches zero, causing the magnitude of the fraction to increase without bound.
5. Step 5: Conclude the behavior of the limit. Since the function becomes increasingly negative as x approaches -2 from the left, the limit is negative infinity.