Infinite Limits


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


b. lim x→0⁻ 2 / (3x¹/³)

Verified step by step guidance

1. Identify the type of limit: This is a one-sided limit as x approaches 0 from the left (x → 0⁻).
2. Analyze the expression: The function is \( \frac{2}{3x^{1/3}} \). The cube root function \( x^{1/3} \) is defined for all real numbers, including negative values.
3. Consider the behavior of \( x^{1/3} \) as x approaches 0 from the left: As x approaches 0 from the negative side, \( x^{1/3} \) approaches 0 from the negative side as well.
4. Evaluate the limit: Since the denominator \( 3x^{1/3} \) approaches 0 from the negative side, the fraction \( \frac{2}{3x^{1/3}} \) will approach negative infinity.
5. Conclude the limit: Therefore, the limit is \( -\infty \).

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. Limits can be finite or infinite, and they are essential for defining derivatives and integrals.

One-Sided Limits

One-sided limits refer to the value that a function approaches as the input approaches a specific point from one side only, either from the left (denoted as x→a⁻) or from the right (denoted as x→a⁺). This concept is crucial when dealing with functions that may behave differently on either side of a point, particularly at points of discontinuity or where the function is not defined.

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 −∞. Understanding infinite limits is important for analyzing the behavior of functions near vertical asymptotes or points where the function tends to grow indefinitely.