Problem 3 - Calculus with Concepts in Calculus

Question

Use the results of this section to evaluate the limit. $$ \lim _{y \rightarrow 64}(\sqrt[3]{y}+\sqrt{y})^{2} $$

Step-by-Step Solution

Answer

The limit is 144.

Step 1: Understand the Problem

We need to find the limit of the given expression as \( y \) approaches 64. The expression is \((\sqrt[3]{y} + \sqrt{y})^2\). This involves both cube roots and square roots of \( y \).

Step 2: Substitute with Limit Value

Start by substituting \( y = 64 \) into the expression to simplify calculations: \( \sqrt[3]{64} + \sqrt{64} \).

Step 3: Evaluate Cube Root and Square Root

Find the cube root of 64: \( \sqrt[3]{64} = 4 \) because \( 4^3 = 64 \). Find the square root of 64: \( \sqrt{64} = 8 \) because \( 8^2 = 64 \).

Step 4: Add the Results

Add the results from the previous steps: \( 4 + 8 = 12 \).

Step 5: Square the Sum

Finally, square the sum from the previous step: \( (12)^2 = 144 \).

Step 6: Conclusion

Therefore, the value of the limit \( \lim_{y \rightarrow 64}(\sqrt[3]{y} + \sqrt{y})^{2} \) is 144.

Key Concepts

Cube Roots

Cube roots are mathematical expressions used to determine a number which, when multiplied by itself three times, gives the original number. For instance, the cube root of 64 is 4, because multiplying 4 by itself three times (i.e., \(4 \times 4 \times 4\)) results in 64.
This operation is denoted as \(\sqrt[3]{64} = 4\). Here are some points to remember about cube roots:

• Cube roots can be calculated for both perfect cubes (like 64) and non-perfect cubes.
• The cube root of a number can be either positive or negative because \((-4)^3\) is also 64.
• Understanding cube roots is essential in solving limit problems involving cubic expressions.

Square Roots

A square root finds a number that, when multiplied by itself, equals the given number. For example, the square root of 64 is 8, as \(8 \times 8 = 64\). This operation is written as \(\sqrt{64} = 8\). Square roots are particularly useful for simplifying and solving equations:

• The square root of a perfect square is always a whole number.
• Unlike cube roots, square roots generally have positive and negative values (\(\pm 8\) for \(\sqrt{64}\)). However, the positive value is typically used in basic arithmetic.
• Square roots play a crucial role in the expression \((\sqrt[3]{y} + \sqrt{y})^2\) when evaluating limits.

Evaluating Limits

Evaluating limits involves finding the value that a function approaches as the input approaches a specific number. In the context of our exercise, we are determining what value \( (\sqrt[3]{y} + \sqrt{y})^2 \) approaches as \( y \) approaches 64.
When evaluating such limits, key steps include:

• Substituting the approaching value into the expression to simplify the function.
• Applying algebraic simplifications as required, such as combining like terms.
• Checking for continuity — ensuring the function behaves predictably around the approaching value.

In this exercise, substituting \( y = 64 \) directly allows us to simplify and conclude efficiently.

Substitution Method

The substitution method in calculus is a straightforward way of evaluating limits by directly replacing the variable with the value it is approaching. In this exercise, we substitute \( y = 64 \) into the expression \((\sqrt[3]{y} + \sqrt{y})^2\). This technique simplifies calculations and is applicable to functions that are continuous at the point of substitution.
Here’s why substitution is useful:

• It reduces complex expressions to simpler forms by eliminating the variable directly.
• It’s especially useful when the limit leads to a simple result without undefined behavior such as division by zero.
• Substitution is quick and efficient when dealing with polynomials and root functions.

Always ensure the substitution doesn't cause division by zero or involve an indeterminate form, such as \(0/0\), requiring other techniques like L'Hôpital's Rule.