Problem 14 - 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 \(c\). $$ \begin{aligned} &\lim _{x \rightarrow c} f(x)=\frac{3}{2} \\ &\lim _{x \rightarrow c} g(x)=\frac{1}{2} \end{aligned} $$

Step-by-Step Solution

Answer

The limits are for (a) 2, (b) 3/4, and (c) 3

Step 1: Find the limit of \(f(x) + g(x)\)

Apply the limit law stating that the limit of a sum is the sum of the limits: \(\lim _{x \rightarrow c}(f(x) + g(x)) = \lim _{x \rightarrow c} f(x) + \lim _{x \rightarrow c} g(x) = \frac {3}{2} + \frac{1}{2} = 2\)

Step 2: Find the limit of \(f(x) * g(x)\)

Apply the limit law stating that the limit of a product is the product of the limits:\(\lim _{x \rightarrow c} (f(x) * g(x)) = \lim _{x \rightarrow c} f(x) * \lim _{x \rightarrow c} g(x) = \frac {3}{2} * \frac{1}{2} = \frac{3}{4}\)

Step 3: Find the Limit of \(f(x) / g(x)\)

Apply the limit law stating that the limit of a quotient is the quotient of the limits, as long as the limit of the denominator is not zero:\(\lim _{x \rightarrow c} (f(x) / g(x)) = \lim _{x \rightarrow c} f(x) / \lim _{x \rightarrow c} g(x) = \frac{3}{2} / \frac{1}{2} = 3\)

Key Concepts

Sum of Limits

When you have two functions, say \(f(x)\) and \(g(x)\), and you are interested in finding the limit as \(x\) approaches a certain number \(c\), the "sum of limits" says that you can simply add their individual limits. This is a huge time-saver and is backed by the limit sum law.

Here's how it works:

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

For example, if \(\lim_{x \rightarrow c} f(x)=\frac{3}{2}\) and \(\lim_{x \rightarrow c} g(x)=\frac{1}{2}\), using the sum of limits, \(\lim_{x \rightarrow c}(f(x) + g(x))\) is \(\frac{3}{2} + \frac{1}{2} = 2\).

So remember, for the sum of limits: just add them up! It's as simple as that.

Product of Limits

The product of limits will help you find the limit of the product of two functions as \(x\) approaches a certain value, say \(c\). This is done using the limit product law which is straightforward and easy to apply.

Here are the steps to use this rule:

• Calculate the limit of \(f(x)\) as \(x\) approaches \(c\).
• Calculate the limit of \(g(x)\) as \(x\) approaches \(c\).
• Multiply these two limits to find the limit of \(f(x) \cdot g(x)\) as \(x\) approaches \(c\).

Using the given function limits, if \(\lim_{x \rightarrow c} f(x)=\frac{3}{2}\) and \(\lim_{x \rightarrow c} g(x)=\frac{1}{2}\), the limit of the product \(\lim_{x \rightarrow c} (f(x) \cdot g(x))\) is \(\frac{3}{2} \cdot \frac{1}{2} = \frac{3}{4}\).

This approach simplifies finding limits by treating the products just like ordinary numbers to multiply. It's effective and efficient!

Quotient of Limits

When dealing with quotients, that is, dividing one function by another, you use the "quotient of limits" rule to find the limit as \(x\) approaches a number \(c\). Be careful with this one because it involves dividing, and division by zero is a no-no!

Here's how to correctly apply this rule:

• Find the limit of \(f(x)\) as \(x\) approaches \(c\).
• Find the limit of \(g(x)\) as \(x\) approaches \(c\).
• If the limit of \(g(x)\) is not zero, divide the limit of \(f(x)\) by the limit of \(g(x)\) to get the limit of \(\frac{f(x)}{g(x)}\) as \(x\) approaches \(c\).

Given \(\lim_{x \rightarrow c} f(x)=\frac{3}{2}\) and \(\lim_{x \rightarrow c} g(x)=\frac{1}{2}\), the quotient of the limits is \(\frac{3}{2} / \frac{1}{2} = 3\).

So, as long as the limit of your denominator isn't zero, you can just divide as normal. If it were zero, that's a different ball game requiring more advanced tools like L'Hôpital's Rule.