Question

Evaluating Limits In Exercises \(37-40,\) use the information to evaluate the limits. $$ \begin{array}{l}{\lim _{x \rightarrow c} f(x)=2} \\ {\lim _{x \rightarrow c} g(x)=\frac{3}{4}} \\ {\text { (a) } \lim _{x \rightarrow c}[4 f(x)]} \\\ {\text { (b) } \lim _{x \rightarrow c}[f(x)+g(x)]} \\ {\text { (c) } \lim _{x \rightarrow c}[f(x) g(x)]} \\ {\text { (d) } \lim _{x \rightarrow c} \frac{f(x)}{g(x)}}\end{array} $$

Short answer

The values of the limits when \(x \rightarrow c\) are: (a) 8, (b) \(\frac{11}{4}\), (c) \(\frac{3}{2}\), (d) \(\frac{8}{3}\).

Step-by-step solution

Evaluate the limit of 4f(x) as x approaches c

Given the limit \(\lim _{x \rightarrow c} f(x) = 2\), apply the constant multiple rule of limits, which states that 'the limit of a constant times a function equals the constant times the limit of the function.' So, \(\lim _{x \rightarrow c}[4 f(x)] = 4 * \lim _{x \rightarrow c} f(x) = 4 * 2 = 8.\)

Evaluate the limit of f(x)+g(x) as x approaches c

Given the limits \(\lim _{x \rightarrow c} f(x) = 2\) and \(\lim _{x \rightarrow c} g(x) = \frac{3}{4}\), apply the sum rule of limits, which states that 'the limit of a sum of functions equals the sum of the limits of the functions.' Therefore, \(\lim _{x \rightarrow c}[f(x) + g(x)] = \lim _{x \rightarrow c} f(x) + \lim _{x \rightarrow c} g(x) = 2 + \frac{3}{4} = \frac{11}{4}.\)

Evaluate the limit of f(x)g(x) as x approaches c

Again using the given limits \(\lim _{x \rightarrow c} f(x) = 2\) and \(\lim _{x \rightarrow c} g(x) = \frac{3}{4}\), apply the product rule of limits, which states that 'the limit of a product of functions equals the product of the limits of the functions.' So, \(\lim _{x \rightarrow c}[f(x) g(x)] = \lim _{x \rightarrow c} f(x) \times \lim _{x \rightarrow c} g(x) = 2 \times \frac{3}{4} = \frac{3}{2}.\)

Evaluate the limit of f(x)/g(x) as x approaches c

Using the provided limits \(\lim _{x \rightarrow c} f(x) = 2\) and \(\lim _{x \rightarrow c} g(x) = \frac{3}{4}\), apply the quotient rule of limits, which states that 'the limit of a quotient of functions equals the quotient of the limits of the functions' (given that the limit of the denominator is not equal to zero). Thus, \(\lim _{x \rightarrow c} \frac{f(x)}{g(x)} = \frac{\lim _{x \rightarrow c} f(x)}{\lim _{x \rightarrow c} g(x)} = \frac{2}{\frac{3}{4}} = \frac{8}{3}.\)

Key concepts

Constant Multiple Rule

The constant multiple rule of limits is a handy tool when dealing with limits in calculus. It simplifies the process of finding the limit of a function multiplied by a constant. The principle is straightforward: when you have a constant multiplied by a function, you can pull the constant out of the limit expression.
The rule says that the limit of a constant multiplied by a function is the constant multiplied by the limit of the function. So, if you know that \(\lim_{x \rightarrow c} f(x) = L\) and you have a constant \(k\), then:

• \(\lim_{x \rightarrow c} [k \cdot f(x)] = k \cdot L\)

This makes evaluating limits much simpler, as you only need to calculate the limit of the function, multiply it by the constant, and you're done! For example, with \(\lim_{x \rightarrow c} f(x) = 2\), if you need to find \(\lim_{x \rightarrow c} [4f(x)]\), you simply multiply the constant 4 by the limit of \(f(x)\), which is 2, resulting in 8.

Sum Rule of Limits

The sum rule of limits helps when you are dealing with the sum of two functions. This rule streamlines the process of evaluating limits for sums. According to this rule, the limit of the sum of two functions is the sum of their individual limits. It's expressed as:

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

This makes it clear that you can evaluate the limits of the individual functions separately and then simply add the results together. Given \(\lim_{x \rightarrow c} f(x) = 2\) and \(\lim_{x \rightarrow c} g(x) = \frac{3}{4}\), you'll find \(\lim_{x \rightarrow c} [f(x) + g(x)]\) by adding these values together:

• \(2 + \frac{3}{4} = \frac{11}{4}\)

Breaking it down this way simplifies the whole process and removes any mystery about how to approach sums in limits.

Product Rule of Limits

The product rule of limits is essential when you want to compute the limit of a product of two functions. This rule states that the limit of the product is simply the product of their limits, which can be expressed as:

• \(\lim_{x \rightarrow c} [f(x) \cdot g(x)] = \lim_{x \rightarrow c} f(x) \cdot \lim_{x \rightarrow c} g(x)\)

This is incredibly useful as it allows you to handle each function's limit individually before multiplying the results.For instance, with \(\lim_{x \rightarrow c} f(x) = 2\) and \(\lim_{x \rightarrow c} g(x) = \frac{3}{4}\), you can easily determine that \(\lim_{x \rightarrow c} [f(x) \cdot g(x)]\) is:

• \(2 \cdot \frac{3}{4} = \frac{3}{2}\)

This approach ensures that evaluating the limit of a product becomes straightforward and less nerve-wracking.

Quotient Rule of Limits

When working with limits involving quotients, the quotient rule provides a simple framework. According to the quotient rule of limits, the limit of a quotient of two functions is the quotient of their limits, as long as the limit of the divisor isn’t zero. Mathematically, this is represented as:

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

Remember, the critical condition is that the limit of \(g(x)\), the denominator, must not be zero, ensuring the division is valid.With given limits \(\lim_{x \rightarrow c} f(x) = 2\) and \(\lim_{x \rightarrow c} g(x) = \frac{3}{4}\), calculate:

• \(\lim_{x \rightarrow c} \frac{f(x)}{g(x)} = \frac{2}{\frac{3}{4}} = \frac{8}{3}\)

This rule is invaluable for keeping quotient limit evaluations clear and manageable, making it much less daunting to handle such problems.