Problem 13 - College Algebra and Calculus: An Applied Approach

Question

Find the limit of (a) \(f(x)+g(x)\), (b) \(f(x) g(x)\), and (c) \(f(x) / g(x)\), as \(x\) approaches \(c\). $$ \begin{aligned} &\lim _{x \rightarrow c} f(x)=3 \\ &\lim _{x \rightarrow c} g(x)=9 \end{aligned} $$

Step-by-Step Solution

Answer

The limits are (a) 12, (b) 27, and (c) 1/3.

Step 1: Calculate Limit of sum

The limit of the sum of two functions is equal to the sum of their limits. In this case, this gives us \(\lim _{x \rightarrow c} f(x)+g(x)= \lim _{x \rightarrow c} f(x) + \lim _{x \rightarrow c} g(x) = 3 + 9 = 12.\)

Step 2: Calculate Limit of Product

The limit of the product of two functions is the product of their limits. So, \(\lim _{x \rightarrow c} f(x)g(x)= \lim _{x \rightarrow c} f(x) \times \lim _{x \rightarrow c} g(x) = 3 \times 9 = 27.\)

Step 3: Calculate Limit of Quotient

The limit of the quotient of two functions is given by the quotient of their limits. In this case, that results in \(\lim _{x \rightarrow c} f(x)/g(x)= \lim _{x \rightarrow c} f(x) / \lim _{x \rightarrow c} g(x)=3 / 9 = 1/3.\) Please note that this operation is valid as long as the limit of the denominator is not zero.

Key Concepts

Limit of a sum

When we talk about the limit of a sum, we refer to finding the limit of the sum of two functions, let's call them \( f(x) \) and \( g(x) \). The limit of a sum rule is very intuitive and simply states that the limit of the sum of two functions is the sum of their individual limits. This is super handy because it lets us deal with complex expressions quite easily. For example, when we have \( \lim_{x \rightarrow c} (f(x) + g(x)) \), we can break it down into:

• First, find the limit of \( f(x) \) as \( x \) approaches \( c \).
• Then, find the limit of \( g(x) \) as \( x \) approaches \( c \).
• Finally, add these two limits together.

So, if \( \lim_{x \rightarrow c} f(x) = 3 \) and \( \lim_{x \rightarrow c} g(x) = 9 \), then the limit of their sum as \( x \) approaches \( c \) is: \( 3 + 9 = 12 \). Using this simple rule helps to simplify a lot of calculus problems and provides a clearer path to finding solutions.

Limit of a product

The limit of a product addresses the situation when we need to determine the limit of a product of two functions, \( f(x) \) and \( g(x) \). Just like the limit of a sum, finding the limit of a product is also quite straightforward with the right approach. The rule here states that the limit of a product of two functions is equivalent to the product of their limits. This offers a systematic way to tackle these limits without overcomplicating things. Let's break it down.

• First, compute the limit of \( f(x) \) as \( x \) approaches \( c \).
• Second, calculate the limit of \( g(x) \) as \( x \) approaches \( c \).
• Multiply these two results together.

Using our earlier examples where \( \lim_{x \rightarrow c} f(x) = 3 \) and \( \lim_{x \rightarrow c} g(x) = 9 \), we find that the limit of their product is: \( 3 \times 9 = 27 \). This process illustrates how knowing just a couple of rules can help navigate through more complex-looking problems with ease.

Limit of a quotient

The concept of the limit of a quotient focuses on determining the limit of one function divided by another function. To find the limit of a quotient, we use a simple yet powerful rule: the limit of a quotient is the quotient of their limits, but this works under certain conditions. Specifically, the rule applies as long as the limit of the denominator is not zero. Let's look at how this rule plays out step by step:

• First, identify the limit of the numerator, \( f(x) \), as \( x \) approaches \( c \).
• Second, determine the limit of the denominator, \( g(x) \), as \( x \) approaches \( c \).
• Finally, divide the limit of the numerator by the limit of the denominator.

In our example, with \( \lim_{x \rightarrow c} f(x) = 3 \) and \( \lim_{x \rightarrow c} g(x) = 9 \), the limit of the quotient \( \frac{f(x)}{g(x)} \) as \( x \) approaches \( c \) is: \( \frac{3}{9} = \frac{1}{3} \). Remember, this procedure only works when the denominator's limit is non-zero, ensuring the quotient can exist under these conditions.