Problem 9 - A Graphical Approach to Precalculus with Limits

Question

Let \(\lim _{x \rightarrow 4} f(x)=16\) and \(\lim _{x \rightarrow 4} g(x)=8 .\) Use the limit rules to find each limit. Do not use a calculator. $$\lim _{x \rightarrow 4} \frac{f(x)+g(x)}{2 g(x)}$$

Step-by-Step Solution

Answer

The limit is \( \frac{3}{2} \).

Step 1: Identify Given Information

We are given that \( \lim_{x \to 4} f(x) = 16 \) and \( \lim_{x \to 4} g(x) = 8 \). We need to find \( \lim_{x \to 4} \frac{f(x) + g(x)}{2g(x)} \).

Step 2: Apply Limit Sum Rule

By the limit laws, the limit of a sum is the sum of the limits. Therefore,\[ \lim_{x \to 4} (f(x) + g(x)) = \lim_{x \to 4} f(x) + \lim_{x \to 4} g(x) = 16 + 8 = 24. \]

Step 3: Apply Limit of a Product Rule

By the limit laws, the limit of a constant times a function is the constant times the limit of the function. Therefore,\[ \lim_{x \to 4} 2g(x) = 2 \cdot \lim_{x \to 4} g(x) = 2 \cdot 8 = 16. \]

Step 4: Apply Limit of a Quotient Rule

The limit of a quotient is the quotient of the limits, given the limit of the denominator is not zero. Therefore,\[ \lim_{x \to 4} \frac{f(x) + g(x)}{2g(x)} = \frac{\lim_{x \to 4} (f(x) + g(x))}{\lim_{x \to 4} 2g(x)} = \frac{24}{16} = \frac{3}{2}. \]

Key Concepts

Sum Rule for Limits

When dealing with limits, the sum rule is a fundamental concept that simplifies the process of finding the limit of sums of functions. If you have two functions, say \( f(x) \) and \( g(x) \), you can determine the limit of their sum as follows. The sum rule states that the limit of a sum is simply the sum of the limits. This means that:

• If \( \lim_{x \to a} f(x) = L \) and \( \lim_{x \to a} g(x) = M \), then \( \lim_{x \to a} [f(x) + g(x)] = L + M \).

So, in the exercise, since \( \lim_{x \to 4} f(x) = 16 \) and \( \lim_{x \to 4} g(x) = 8 \), by applying the sum rule, we can simply add these two limits together:

• \( \lim_{x \to 4} (f(x) + g(x)) = 16 + 8 = 24 \).

This rule greatly simplifies the process and ensures accuracy when dealing with limits of sums.

Product Rule for Limits

The product rule for limits is another essential tool that helps find the limit of a product of functions. According to this rule, if you know the limits of individual functions, you can find the limit of their product. The rule states that the limit of a product is the product of the limits:

• Suppose \( \lim_{x \to a} f(x) = L \) and \( \lim_{x \to a} g(x) = M \), then \( \lim_{x \to a} [f(x) \cdot g(x)] = L \cdot M \).

In the exercise, we deal specifically with the limit of \( 2g(x) \). Here, \( 2 \) is a constant, and the product rule simplifies to a special case involving constants:

• \( \lim_{x \to 4} 2g(x) = 2 \cdot \lim_{x \to 4} g(x) = 2 \cdot 8 = 16 \).

This rule makes it easy to handle limits involving products, especially when constants are involved.

Quotient Rule for Limits

The quotient rule for limits is crucial for finding the limit of quotients of functions. It is applied when you have one function divided by another. The quotient rule asserts that the limit of a quotient is the quotient of the limits, provided the denominator limit is not zero:

• If \( \lim_{x \to a} f(x) = L \) and \( \lim_{x \to a} g(x) = M \), with \( M eq 0 \), then \( \lim_{x \to a} \frac{f(x)}{g(x)} = \frac{L}{M} \).

In this exercise, we need to find \( \lim_{x \to 4} \frac{f(x) + g(x)}{2g(x)} \). By the quotient rule,

• We first found \( \lim_{x \to 4} (f(x) + g(x)) = 24 \) and \( \lim_{x \to 4} 2g(x) = 16 \).
• Thus, \( \lim_{x \to 4} \frac{f(x) + g(x)}{2g(x)} = \frac{24}{16} = \frac{3}{2} \).

The quotient rule is particularly useful and ensures the limit process is straightforward, as long as the denominator limit is non-zero.