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Magazine Chapter 2: Problem 3
Use the results of this section to evaluate the limit. $$ \lim _{y \rightarrow 64}(\sqrt[3]{y}+\sqrt{y})^{2} $$
Short Answer
Expert verified
The limit is 144.
Step by step solution
01
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 \( y \).
02
Substitute with Limit Value
Start by substituting \( y = 64 \) into the expression to simplify calculations: \( \sqrt[3]{64} + \sqrt{64} \).
03
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 \( 8^2 = 64 \).
04
Add the Results
Add the results from the previous steps: \( 4 + 8 = 12 \).
05
Square the Sum
Finally, square the sum from the previous step: \( (12)^2 = 144 \).
06
Conclusion
Therefore, the value of the limit \( \lim_{y \rightarrow 64}(\sqrt[3]{y} + \sqrt{y})^{2} \) is 144.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
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:
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:
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 鈥� ensuring the function behaves predictably around the approaching value.
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鈥檚 why substitution is useful:
Here鈥檚 why substitution is useful:
- It reduces complex expressions to simpler forms by eliminating the variable directly.
- It鈥檚 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.
