Problem 4 - Calculus with Concepts in Calculus

Question

Use the results of this section to evaluate the limit. $$ \lim _{t \rightarrow 0} \frac{2 t^{1 / 3}-4}{-3 t^{1 / 3}+5} $$

Step-by-Step Solution

Answer

The limit is \( \frac{-4}{5} \).

Step 1: Identify the indeterminate form

As \( t \to 0 \), both the numerator and denominator approach constants that result in \((\cdot-4)/(-3\cdot+5)\). Let's evaluate the expression by substituting \( t = 0 \).

Step 2: Substitute t = 0 directly in the expression

Substitute \( t = 0 \) into the expression: \( \frac{2(0)^{1/3} - 4}{-3(0)^{1/3} + 5} = \frac{-4}{5} \).

Step 3: Evaluate direct substitution

Since the substitution yields \( \frac{-4}{5} \), which is a non-zero number, the expression is evaluated directly without needing further simplification or limit rules.

Key Concepts

Indeterminate Forms

When evaluating limits, we sometimes encounter expressions that seem undefined at a first glance. These are called indeterminate forms and arise particularly in cases such as \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \). Indeterminate forms indicate that more analysis is needed to determine the actual limit.

In the given exercise, the expression as \( t ightarrow 0 \) appears not as one of the classic indeterminate forms at a first evaluation. Instead, the numerator evaluates to a constant \(-4\), and the denominator to a constant \(5\). Since this is not an indeterminate form, evaluating it is straightforward. It's an important first step to always check for indeterminate forms when approaching limits. Doing so guides us on whether to utilize further techniques like L'Hôpital's Rule or algebraic manipulation.

Direct Substitution

Direct substitution is often the simplest way to evaluate a limit. It involves substituting the approaching value directly into the function. If this substitution results in a meaningful, defined number, you've found the limit.

In the exercise, when substituting \( t = 0 \) into the expression \( \frac{2t^{1/3}-4}{-3t^{1/3}+5} \), the cubic roots of zero equal zero, simplifying our expression significantly. This gives us \( \frac{-4}{5} \), a perfectly valid number that does not require further manipulation.

• This method is particularly effective when the result of substitution doesn't result in indeterminate forms like \( \frac{0}{0} \).
• If direct substitution resolves to a determinate result, you can confirm that result as the limit.

Thus, always consider trying direct substitution first, as it often reveals solutions quickly.

Limit Evaluation Steps

Evaluating limits is a stepwise process that guides you toward finding an accurate value. Here's how you would typically proceed:

• **Identify the Form**: Check if the expression yields an indeterminate form like \( \frac{0}{0} \). If not, you might use simpler methods.
• **Attempt Direct Substitution**: Plug in the approaching value into the function. If it yields a finite value, that's your limit.
• **Simplify if Needed**: If direct substitution results in indeterminacy, algebraic manipulation or using limit laws might be necessary.
• **Apply Advanced Techniques as Needed**: If algebraic simplification still results in indeterminacy, advanced techniques such as L'Hôpital’s Rule might be necessary for evaluation.

In this exercise, the steps illustrate the simplicity that can often be found in limit problems. By determining there was no indeterminacy and attempting direct substitution, the limit was evaluated quickly and correctly as \( \frac{-4}{5} \). Thus, clearly going through the correct steps ensures accurate results in limit evaluation.