Problem 57 - Precalculus

Question

Evaluate the limits using the limit properties. $$\lim _{x \rightarrow 10} 5 \sqrt[3]{2 x+7}$$

Step-by-Step Solution

Answer

The evaluated limit is 15.

Step 1: Identify the Separate Components

The function provided is composed of a constant factor (5) and a cube root function, \(\sqrt[3]{2x +7}\). We need to evaluate the limit of each component separately as \(x\) approaches 10.

Step 2: Use the Limit Product Property

According to the limit product property, the limit of a product is the product of the limits (if they exist). Hence, we can rewrite the original limit as:\[\lim_{x \to 10} 5 \cdot \sqrt[3]{2x + 7} = 5 \cdot \lim_{x \to 10} \sqrt[3]{2x + 7}\]

Step 3: Evaluate the Cube Root Component

Evaluate the limit of the cube root component. Substitute \(x = 10\) into the inner expression:\[\lim_{x \to 10} \sqrt[3]{2x + 7} = \sqrt[3]{2(10) + 7} = \sqrt[3]{27} = 3\]The limit of \(\sqrt[3]{2x + 7}\) as \(x\) approaches 10 is 3.

Step 4: Combine the Results

Use the results from Step 2 and Step 3 to find the final answer. Multiply the constant 5 by the result from evaluating the cube root:\[5 \cdot 3 = 15\]Thus, the evaluated limit is 15.

Key Concepts

Constants in Limits

When solving limits, constants play an important role.
Constants are numbers that do not change, regardless of the variable's value. In the exercise, you see the constant 5. It's a factor of the entire expression.
The limit of a constant multiplied by a function is simply the constant multiplied by the limit of the function itself. In the formula:

• If you have a limit, \( \lim_{x \to a} [c \cdot f(x)] \), where \(c\) is a constant.
• You can separate it as \( c \cdot \lim_{x \to a} f(x) \).

This property simplifies calculations since you often handle the function and constant separately. When you encounter limits with a constant, this property is very useful when simplifying complex expressions.

Cube Root Function

The cube root function is symbolized as \(\sqrt[3]{x}\). It is designed to find a number whose cube equals \(x\).
In our exercise, you see it as \(\sqrt[3]{2x + 7}\), which means you are finding the cube root of the expression \(2x + 7\). Cube root functions have a key attribute: they are continuous everywhere. Thus, you can directly substitute \(x\) for almost every value.
This automatic continuity simplifies limit evaluation where direct substitution often provides the limit.
For example, when \(x = 10\), substituting gives:

• First, compute the inside: \(2(10) + 7 = 27\).
• Then, find the cube root: \(\sqrt[3]{27} = 3\).

This direct substitution handles breaking down and solving the expression inside the cube root, as shown in the original step by step solution.

Substitution Method in Limits

The substitution method in limits is a straightforward technique often applied to find a limit without advanced functions complicating the outcome.
This technique involves directly substituting a value into a function to find the limit.
For the function given in the exercise, substituting \(x = 10\) directly into the expression, \(2x + 7\), leads you to the next step in evaluating the cube root.Substitution helps when:

• The function is continuous at the point you're evaluating.
• No undefined expressions or indeterminate forms arise (like \(\frac{0}{0}\)).

It's simple, efficient, and empowers you to find limits reliably, as shown when substituting helped determine the cube root to finish calculating the expression. Learning to use substitution wisely is key in solving many calculus problems efficiently.