Infinite Limits


Find the limits in Exercises 37–48. Write ∞ or −∞ where appropriate.


lim x→0 (−1) / (x² (x + 1))

Verified step by step guidance

1. First, identify the type of limit you are dealing with. Here, you have a limit as x approaches 0 for the function \(-\frac{1}{x^2(x+1)}\).
2. Examine the behavior of the denominator as x approaches 0. Notice that \(x^2\) approaches 0 and \(x+1\) approaches 1, making the denominator approach 0.
3. Since the denominator approaches 0, the fraction \(-\frac{1}{x^2(x+1)}\) will tend towards infinity or negative infinity depending on the sign of the numerator and denominator.
4. Consider the sign of the expression as x approaches 0 from the positive side (x → 0⁺). The denominator \(x^2(x+1)\) is positive, and the numerator is negative, leading to \(-∞\).
5. Similarly, consider the sign of the expression as x approaches 0 from the negative side (x → 0⁻). The denominator \(x^2(x+1)\) is positive, and the numerator is negative, also leading to \(-∞\).

Key Concepts

Limits

A limit is a fundamental concept in calculus that describes the behavior of a function as its input approaches a certain value. It helps in understanding how functions behave near points of interest, including points where they may not be defined. In this case, we are interested in the limit as x approaches 0, which requires analyzing the function's values as x gets closer to 0 from both the left and right.

Infinite Limits

Infinite limits occur when the value of a function increases or decreases without bound as the input approaches a certain point. This can result in the limit being expressed as ∞ or −∞. In the given problem, we need to determine if the function approaches positive or negative infinity as x approaches 0, which involves examining the signs and magnitudes of the function's components.

Behavior of Rational Functions

Rational functions are ratios of polynomials, and their limits can often be analyzed by examining the degrees of the polynomials in the numerator and denominator. In this case, the function involves a polynomial in the denominator that approaches zero as x approaches 0, which can lead to infinite limits. Understanding how the numerator and denominator interact is crucial for determining the limit's value.