Find the limits in Exercises 59–62. Write ∞ or −∞ where appropriate.


lim (2 − 3 / t¹/³) as


a. t → 0⁺

Verified step by step guidance

1. Identify the expression for which you need to find the limit: \( 2 - \frac{3}{t^{1/3}} \).
2. Recognize that as \( t \to 0^+ \), \( t^{1/3} \to 0^+ \) as well, since the cube root of a positive number approaching zero also approaches zero.
3. Consider the behavior of the term \( \frac{3}{t^{1/3}} \). As \( t^{1/3} \to 0^+ \), \( \frac{3}{t^{1/3}} \to +\infty \) because dividing by a very small positive number results in a very large positive number.
4. Analyze the entire expression \( 2 - \frac{3}{t^{1/3}} \). Since \( \frac{3}{t^{1/3}} \to +\infty \), the expression \( 2 - \frac{3}{t^{1/3}} \) will tend towards \(-\infty\).
5. Conclude that the limit of the expression as \( t \to 0^+ \) is \(-\infty\).

Key Concepts

Limits

Limits are fundamental concepts in calculus that describe the behavior of a function as its input approaches a certain value. They help in understanding how functions behave near specific points, including points of discontinuity or infinity. In this case, we are interested in the limit of the function as t approaches 0 from the positive side (0⁺).

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 the left (−) or the right (+). In this question, we are evaluating the right-hand limit as t approaches 0, which is crucial for determining the function's behavior in that region without considering values from the left.

Cube Root Function

The cube root function, denoted as t¹/³, is a continuous function that returns the number which, when cubed, gives the input value. As t approaches 0, the cube root of t also approaches 0. Understanding how this function behaves near 0 is essential for evaluating the limit in the given problem.