Question

Evaluating Limits In Exercises \(37-40,\) use the information to evaluate the limits. $$ \begin{array}{l}{\lim _{x \rightarrow c} f(x)=27} \\ {\text { (a) } \lim _{x \rightarrow c} \sqrt[3]{f(x)}} \\ {\text { (b) } \lim _{x \rightarrow c} \frac{f(x)}{18}} \\ {\text { (c) } \lim _{x \rightarrow c}[f(x)]^{2}} \\\ {\text { (d) } \lim _{x \rightarrow c}[f(x)]^{2 / 3}}\end{array} $$

Short answer

The solutions to the limits are: (a) 3, (b) 1.5, (c) 729, and (d) 9.

Step-by-step solution

Evaluate limit (a)

Using the limit property that the limit of a function raised to a power is equal to the value raised to that power, we can solve for the limit. So, we have \(\lim_{x \rightarrow c} \sqrt[3]{f(x)} = \sqrt[3]{\lim_{x \rightarrow c} f(x)} = \sqrt[3]{27} = 3\).

Evaluate limit (b)

Using the limit property that the limit of a function times a constant is equal to the actual limit times that constant, we find \(\lim_{x \rightarrow c} \frac{f(x)}{18} = \frac{\lim_{x \rightarrow c} f(x)}{18} = \frac{27}{18} = 1.5\).

Evaluate limit (c)

Once again, we use the limit property that the limit of a function raised to a power is equal to the value raised to that power: \(\lim_{x \rightarrow c} [f(x)]^{2} = (\lim_{x \rightarrow c} f(x))^{2} = 27^{2} = 729\).

Evaluate limit (d)

For the last limit, we use the same property as in the former steps. Hence, \(\lim_{x \rightarrow c} [f(x)]^{2 / 3} = (\lim_{x \rightarrow c} f(x))^{2 / 3} = 27^{2 / 3} = 9\).

Key concepts

Limit Properties

Limits are fundamental in calculus as they describe the behavior of functions as they approach a certain point. The properties of limits are vital for solving more complex problems.

Some essential limit properties you should know include:

• Sum Rule: The limit of a sum is the sum of the limits. If \(\lim _{x \rightarrow c} f(x) = L\) and \(\lim _{x \rightarrow c} g(x) = M\), then \(\lim _{x \rightarrow c} [f(x) + g(x)] = L + M\).


• Product Rule: The limit of a product is the product of the limits. If \(\lim _{x \rightarrow c} f(x) = L\) and \(\lim _{x \rightarrow c} g(x) = M\), then \(\lim _{x \rightarrow c} [f(x) \cdot g(x)] = L \cdot M\).


• Quotient Rule: The limit of a quotient is the quotient of the limits, provided the limit of the denominator is not zero. It states that \(\lim _{x \rightarrow c} [\frac{f(x)}{g(x)}] = \frac{L}{M}\), where \(M eq 0\).

These properties simplify the process so you can break down complicated expressions into manageable parts.

Function Limits

Understanding function limits helps predict the behavior of functions at particular points. It offers a way to find a value for expressions where direct substitution could result in indeterminacy.

For instance, consider the limit problem \(\lim_{x \rightarrow c} f(x) = 27\). This tells us that as \(x\) gets closer to \(c\), \(f(x)\) approaches 27. Determining function limits is about finding that approaching value by analyzing a function's behavior around that particular 'c'.

When dealing with limits, we also utilize left-hand and right-hand limits. This means examining the behavior of the function from both sides of the point in question to ensure continuity and to confirm the limit exists.

• If both the left-hand and right-hand limits are equal, then the limit at that point is said to exist.

Thus, Function limits form a cornerstone for understanding broader concepts in calculus.

Power of a Function

Raising a function to a power involves limits that follow a neat rule: the limit of a function raised to a power equals that limit raised to that power. This simplifies calculations significantly.

In our original problem, we see applications of this rule. For example:

• The limit of a cube root, \(\lim_{x \rightarrow c} \sqrt[3]{f(x)} = \sqrt[3]{\lim_{x \rightarrow c} f(x)}\). By evaluating \(\lim_{x \rightarrow c} f(x) = 27\), it results in \(\sqrt[3]{27} = 3\).
• The limit of a squared function, \(\lim_{x \rightarrow c} [f(x)]^{2} = (\lim_{x \rightarrow c} f(x))^{2} = 27^{2} = 729\).

This rule is incredibly useful as it means you can first evaluate the basic limit, then apply the power, saving you complicated steps and calculations.

Constant Multiplication in Limits

When we multiply a function by a constant and then take the limit, it is straightforward. The limit of a function multiplied by a constant equals the constant multiplied by the limit of the function itself.

Let's see an example from the solution:

• Consider the problem \(\lim_{x \rightarrow c} \frac{f(x)}{18}\). This simplifies to \(\frac{1}{18} \cdot \lim_{x \rightarrow c} f(x)\). With \(\lim_{x \rightarrow c} f(x) = 27\), it becomes \(\frac{27}{18} = 1.5\).

The property of constant multiplication ensures that constants move outside the limit operation, making calculations simpler and less error-prone.

It's an essential skill when approaching more complex limit expressions. Applying this property decreases the complexity of problems and expands your toolkit for handling a variety of limit scenarios.