Problem 4 - Contemporary Precalculus

Question

Use the following facts about the functions f, g, and h to find the required limit. $$\begin{aligned} &\lim _{x \rightarrow 4} f(x)=5 \quad \lim _{x \rightarrow 4} g(x)=0\\\ &\lim _{x \rightarrow 4} h(x)=-2 \end{aligned}$$ $$\lim _{x \rightarrow 4} \frac{g(x)}{h(x)}$$

Step-by-Step Solution

Answer

Question: What is the limit of g(x)/h(x) as x approaches 4, given the following limit values: $$\begin{aligned} &\lim_{x \rightarrow 4} f(x) = 5\\ &\lim_{x \rightarrow 4} g(x) = 0\\ &\lim_{x \rightarrow 4} h(x) = -2 \end{aligned}$$ Answer: The limit of g(x)/h(x) as x approaches 4 is 0.

Step 1: Write Down the Given Information

We are given the following limit values as x approaches 4: $$\begin{aligned} &\lim_{x \rightarrow 4} f(x) = 5\\ &\lim_{x \rightarrow 4} g(x) = 0\\ &\lim_{x \rightarrow 4} h(x) = -2 \end{aligned}$$ We want to find: $$\lim_{x \rightarrow 4} \frac{g(x)}{h(x)}$$

Step 2: Apply the Limit Properties

To find the limit of the quotient g(x)/h(x) as x approaches 4, we can use the quotient limit rule: $$\lim_{x \rightarrow a} \frac{f(x)}{g(x)} = \frac{\lim_{x \rightarrow a} f(x)}{\lim_{x \rightarrow a} g(x)}$$ So, we have: $$\lim_{x \rightarrow 4} \frac{g(x)}{h(x)} = \frac{\lim_{x \rightarrow 4} g(x)}{\lim_{x \rightarrow 4} h(x)}$$

Step 3: Plug in the Given Limit Values

Now, we can substitute the given limit values for g(x) and h(x) into the expression: $$\frac{\lim_{x \rightarrow 4} g(x)}{\lim_{x \rightarrow 4} h(x)} = \frac{0}{-2}$$

Step 4: Evaluate and Simplify

Now, we can simplify the expression to find the limit of g(x)/h(x) as x approaches 4: $$\frac{0}{-2} = 0$$ So, the limit of g(x)/h(x) as x approaches 4 is 0: $$\lim_{x \rightarrow 4} \frac{g(x)}{h(x)} = 0$$

Key Concepts

The Quotient Limit Rule

Understanding the quotient limit rule is essential when working with limits involving division in precalculus. Simply stated, the quotient limit rule allows us to find the limit of a quotient of two functions by dividing the limit of the numerator by the limit of the denominator, provided the limit of the denominator is not zero.

For example, if you're given two functions, say, \(f(x)\) and \(g(x)\), and you need to find \(\lim_{x \rightarrow a} \frac{f(x)}{g(x)}\), you can use this rule to break it down into \(\frac{\lim_{x \rightarrow a} f(x)}{\lim_{x \rightarrow a} g(x)}\) assuming \(\lim_{x \rightarrow a} g(x) \eq 0\). In the exercise provided, we applied the quotient limit rule to the functions \(g(x)\) and \(h(x)\) as \(x\) approaches 4, which simplified the problem significantly.

Thus, whenever you are faced with a limit of a quotient, remember to check if you can apply this rule. It significantly simplifies complex problems and is one of the foundational tools in your precalculus toolkit.

Finding Limits Algebraically

Finding limits algebraically is a key skill that involves manipulating expressions to determine what value they approach as the variable approaches a certain point. Algebraic techniques include factoring, combining like terms, rationalizing, and using limit rules such as the quotient limit rule.

In the exercise above, the process is straightforward as we directly substitute values using the known limits. But, often the function can be more complicated, and direct substitution isn't possible due to indeterminate forms like \(0/0\). In such cases, algebraic manipulation is critical to simplify the expression or eliminate the indeterminate form to find the limit.

Students should become comfortable with these algebraic manipulations, as they are not only useful in finding limits but also serve as foundational skills for calculus—where understanding the behavior of functions as they approach a specific value is often required.

Limit Properties

Limit properties are the rules that allow us to perform operations on limits, and these rules make solving limit problems much easier. Important properties include the sum, difference, product, quotient, and power rules. These properties are based on the behavior of limits and can be used to find the limits of more complex expressions based on the limits of simpler ones.

For instance, the sum rule says that the limit of a sum is the sum of the limits, while the quotient rule, which is particularly relevant for our exercise, indicates that the limit of a quotient is the quotient of the individual limits, as long as the limit of the denominator isn't zero. Employing these properties, as we did earlier, allows us to take the given limits of \(g(x)\) and \(h(x)\) and combine them to find the limit of \(g(x)/h(x)\).

Understanding and using limit properties is crucial to not only evaluating limits but also to developing a deeper understanding of function behavior, continuity, and the fundamental concepts that underpin calculus.