Problem 15 - College Algebra and Calculus: An Applied Approach

Question

Find the limit of (a) \(\sqrt{f(x)}\), (b) \([3 f(x)]\), and (c) \([f(x)]^{2}\), as \(x\) approaches \(c\). $$ \lim _{x \rightarrow c} f(x)=16 $$

Step-by-Step Solution

Answer

The limit of \(\sqrt{f(x)}\) as \(x\) approaches \(c\) is 4. The limit of \([3 f(x)]\) as \(x\) approaches \(c\) is 48. The limit of \([f(x)]^{2}\) as \(x\) approaches \(c\) is 256.

Step 1: Take the Limit of \(\sqrt{f(x)}\)

Applying the rule discussed above, take the limit of \(\sqrt{f(x)}\) as \(x\) approaches \(c\). Given that \(\lim_{x \rightarrow c} f(x) = 16\), we substitute \(f(x) = 16\) into our function: \(\sqrt{f(x)} = \sqrt{16}\).

Step 2: Take the Limit of \([3 f(x)]\)

Applying the rule, the limit of \([3 f(x)]\) as \(x\) approaches \(c\) is obtained by substituting \(f(x) = 16\) into the function: \(3f(x) = 3 \cdot 16\).

Step 3: Take the Limit of \([f(x)]^{2}\)

Applying the rule, the limit of \([f(x)]^{2}\) as \(x\) approaches \(c\) is obtained by substituting \(f(x) = 16\) into the function: \([f(x)]^{2} = 16^{2}\).

Key Concepts

Limit Laws

In calculus, limit laws are fundamental rules that simplify the process of finding limits of various functions. These laws allow us to break down complex expressions into simpler components that are easier to handle.

• **Constant Multiple Law:** This rule states that if you have a constant multiplying a function, you can take the constant out of the limit: \ \ \ \(\lim_{x \to c} [a \cdot f(x)] = a \cdot \lim_{x \to c} f(x)\).
• **Sum/Difference Law:** To find the limit of a sum or difference of functions, take the limit of each function separately: \ \ \ \(\lim_{x \to c} [f(x) \pm g(x)] = \lim_{x \to c} f(x) \pm \lim_{x \to c} g(x)\).
• **Product Law:** The limit of a product is the product of the limits: \ \ \ \(\lim_{x \to c} [f(x) \cdot g(x)] = \lim_{x \to c} f(x) \cdot \lim_{x \to c} g(x)\).
• **Power Law:** If you have a function raised to a power, the limit can be taken first, then raised to the power: \ \ \ \(\lim_{x \to c} [f(x)]^n = \left(\lim_{x \to c} f(x)\right)^n \).

Understanding these basic laws allows you to easily calculate the limit of more complicated expressions, as seen in the exercise above with expressions like \(\sqrt{f(x)}\), \(3f(x)\), and \([f(x)]^2\).

Continuity

Continuity in mathematics, particularly in calculus, refers to the behavior of a function at a point and its immediate vicinity. A function is said to be continuous at a point \(c\) if the following conditions are met:

• The function \(f(x)\) is defined at \(c\): \(f(c)\) exists.
• The limit of \(f(x)\) as \(x\) approaches \(c\) exists: \(\lim_{x \to c} f(x)\) exists.
• The value of the function at \(c\) equals the limit as \(x\) approaches \(c\): \(f(c) = \lim_{x \to c} f(x)\).

This means there's no 'jump' or 'gap' in the function at the point.
Continuity plays a crucial role in calculus because it often simplifies the calculation of limits. For continuous functions, the limit and the value of the function at a point are the same, which means you can directly substitute \(x = c\) into the function to find the limit.

Functions

Functions are the backbone of calculus and are essential for understanding limits. A function \(f(x)\) represents a relationship between a set of inputs and outputs, where each input \(x\) corresponds to exactly one output \(f(x)\).

• **Types of Functions:** Functions can be polynomial, rational, exponential, logarithmic, trigonometric, and more. Each type behaves differently and its limits are calculated using specific rules.
• **Evaluating Functions:** To evaluate a function means finding the output \(f(x)\) for a given input \(x\). This is a key step when applying limit laws to calculate the limit as \(x\) approaches a certain value.
• **Behavior of Functions:** Depending on the type of function, they can behave very differently near certain values. Understanding the behavior is crucial for applying the right limit laws and continuity conditions.

In the exercise, we work with the function \(f(x)\) and calculate limits as \(x\) approaches \(c\). This involves using function evaluations like \(\sqrt{f(x)}\), \(3f(x)\), and \([f(x)]^{2}\), all of which require understanding the nature of \(f(x)\) and its limits.

Approaching a Value

In the context of limits, "approaching a value" refers to how the inputs of a function get infinitely close to a particular number. For a given function \(f(x)\), if \(x\) is approaching a specific value \(c\), we are interested in the behavior of \(f(x)\) as \(x\) moves closer and closer to \(c\) without necessarily reaching it.

• To understand what happens as \(x\) approaches \(c\), we look at the values of \(f(x)\) where \(x\) is near \(c\), and see if these values are settling around a particular number.
• This approach helps resolve cases where directly plugging in \(c\) into \(f(x)\) isn’t possible, like when \(f(x)\) involves division by zero or other undefined operations at \(c\).

In the exercise provided, the problem is to find out what happens to \(f(x)\) and its variants \(\sqrt{f(x)}\), \(3f(x)\), and \([f(x)]^2\) as \(x\) gets close to \(c\). Working through these problems helps build a strong intuition about function behaviors and limit calculation, crucial for mastering calculus.