Question

Evaluating Limits In Exercises \(37-40,\) use the information to evaluate the limits. $$ \begin{array}{l}{\lim _{x \rightarrow c} f(x)=3} \\ {\lim _{x \rightarrow c} g(x)=2} \\ {\text { (a) } \lim _{x \rightarrow c}[5 g(x)]} \\ {\text { (b) } \lim _{x \rightarrow c}[f(x)+g(x)]} \\ {\text { (d) } \lim _{x \rightarrow c}[f(x) g(x)]} \\ {\text { (d) } \lim _{x \rightarrow c} \frac{f(x)}{g(x)}}\end{array} $$

Short answer

The results of the limits are: a) 10, b) 5, c) 6, and d) 1.5.

Step-by-step solution

Evaluate \(\lim_{x \rightarrow c}[5g(x)]\)

Using the property 'Limit of a constant times a function', we know that \(\lim_{x \rightarrow c}[5g(x)] = 5 * \lim_{x \rightarrow c} g(x) = 5 * 2 = 10.\)

Evaluate \(\lim_{x \rightarrow c}[f(x) + g(x)]\)

Using the property 'Sum of limits', we get \(\lim_{x \rightarrow c}[f(x) + g(x)] = \lim_{x \rightarrow c}f(x) + \lim_{x \rightarrow c}g(x) = 3 + 2 = 5.\)

Evaluate \(\lim_{x \rightarrow c}[f(x)g(x)]\)

Using the property 'Product of limits', we obtain \(\lim_{x \rightarrow c}[f(x)g(x)] = \lim_{x \rightarrow c}f(x) * \lim_{x \rightarrow c}g(x) = 3 * 2 = 6.\)

Evaluate \(\lim_{x \rightarrow c}\frac{f(x)}{g(x)}\)

Finally, using the property 'Quotient of Limits' (Given the limit of the denominator is not zero), we find \(\lim_{x \rightarrow c}\frac{f(x)}{g(x)} = \frac{\lim_{x \rightarrow c}f(x)}{\lim_{x \rightarrow c}g(x)} = \frac{3}{2} = 1.5.\)

Key concepts

Limit of a Constant Times a Function

The concept of the "limit of a constant times a function" helps simplify many limit calculations. When you multiply a function by a constant and need to find the limit, you can pull out the constant from the limit expression. This rule makes the computation easier by allowing us to focus on the limit of the function itself rather than worrying about the constant as it approaches the limit.For example, if you have \(5g(x)\), and you know \(\lim_{x \rightarrow c} g(x) = 2\), then

• \(\lim_{x \rightarrow c}[5g(x)] = 5 \cdot \lim_{x \rightarrow c} g(x)\)
• This becomes \(5 \times 2 = 10\)

Breaking it down:

• First, recognize the constant: 5
• Next, calculate the limit of the function: 2
• Finally, multiply the constant by this limit

This method ensures limits are handled efficiently by isolating the contribution of constants.

Sum of Limits

The property "sum of limits" allows us to break down complex expressions into simpler parts and compute their limits individually. If you have two functions, say \(f(x)\) and \(g(x)\), getting their limit as a sum is straightforward. You simply find the limits of each function separately and then add them together.Here's how it works:

• If \(\lim_{x \rightarrow c} f(x) = 3\) and \(\lim_{x \rightarrow c} g(x) = 2\)
• Then \(\lim_{x \rightarrow c} [f(x) + g(x)] = \lim_{x \rightarrow c} f(x) + \lim_{x \rightarrow c} g(x)\)
• This results in \(3 + 2 = 5\)

The steps are simple:

• First, determine the limit of each function separately
• Then, add the two limits together

This approach is particularly helpful as it simplifies the process of limit calculation by treating each function independently.

Product of Limits

The "product of limits" property is a practical approach when dealing with two functions that are being multiplied. This concept allows you to multiply the limits of each function to find the limit of the product as a whole. Imagine multiplying two functions \(f(x)\) and \(g(x)\). If their limits are known, the product of these limits will give you the result you need.

• Given \(\lim_{x \rightarrow c} f(x) = 3\) and \(\lim_{x \rightarrow c} g(x) = 2\)
• Then \(\lim_{x \rightarrow c} [f(x)g(x)] = \lim_{x \rightarrow c} f(x) \cdot \lim_{x \rightarrow c} g(x)\)
• This becomes \(3 \times 2 = 6\)

The steps involve:

• First, calculate the limit of each function separately
• Next, multiply these limits to get your result

This property ensures that product limits can be handled smoothly and accurately by simplifying the multiplication of functions into a more manageable form.

Quotient of Limits

The "quotient of limits" property describes a useful technique for finding the limit of a division between two functions. The limit of the quotient can be determined by dividing the limits of each function, provided that the limit of the denominator is not zero.To apply this property, consider two functions \(f(x)\) and \(g(x)\) where:

• \(\lim_{x \rightarrow c} f(x) = 3\) and \(\lim_{x \rightarrow c} g(x) = 2\)
• Ensure \(\lim_{x \rightarrow c} g(x) eq 0\) to avoid undefined expressions

So,

• \(\lim_{x \rightarrow c} \frac{f(x)}{g(x)} = \frac{\lim_{x \rightarrow c} f(x)}{\lim_{x \rightarrow c} g(x)}\)
• Which equals \(\frac{3}{2} = 1.5\)

The process requires:

• First, compute the limit of the numerator
• Second, compute the limit of the denominator
• Lastly, divide the two results

This property is powerful for safely and effectively evaluating the limits of fractions.