Problem 26 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find the limits if \(\lim _{x \rightarrow a} f(x)=3\) and \(\lim _{x \rightarrow a} g(x)=-1\) (see Example 4\()\). $$\lim _{x \rightarrow a} \frac{2 f(x)-3 g(x)}{f(x)+g(x)}$$

Step-by-Step Solution

Answer

The limit is 4.5.

Step 1: Understand the Problem

We need to find the limit of the function \( \frac{2f(x) - 3g(x)}{f(x) + g(x)} \) as \( x \rightarrow a \). We are given that \( \lim _{x \rightarrow a} f(x)=3 \) and \( \lim _{x \rightarrow a} g(x)=-1 \).

Step 2: Apply Limit Properties

Use the properties of limits: \( \lim_{x \rightarrow a} \left[ 2f(x) - 3g(x) \right] = 2 \lim_{x \rightarrow a} f(x) - 3 \lim_{x \rightarrow a} g(x) \).

Step 3: Calculate the Numerator Limit

Substitute the given limits into the expression: \[2 \times 3 - 3 \times (-1) = 6 + 3 = 9.\]So, the numerator limit is 9.

Step 4: Calculate the Denominator Limit

Substitute the given limits into the expression for the denominator: \[\lim_{x \rightarrow a} \left[ f(x) + g(x) \right] = \lim_{x \rightarrow a} f(x) + \lim_{x \rightarrow a} g(x) = 3 + (-1) = 2.\]So, the denominator limit is 2.

Step 5: Divide the Limits

Finally, divide the limit of the numerator by the limit of the denominator to find the overall limit:\[\frac{9}{2} = 4.5.\]

Step 6: Verify and Summarize

Verify that the calculations are consistent with the limit rules and summarize the solution: The limit of the function as \( x \rightarrow a \) is \( 4.5 \).

Key Concepts

Limit Properties

Limit properties are mathematical rules that help us to calculate the limits of complex functions by breaking them down into simpler parts. By understanding these properties, you can solve problems more efficiently. For any functions \(f(x)\) and \(g(x)\), when \(x\) approaches a constant value \(a\), some key properties are used:

• The sum of the limits: \(\lim_{x\to a} (f(x) + g(x)) = \lim_{x\to a} f(x) + \lim_{x\to a} g(x)\)
• The difference of limits: \(\lim_{x\to a} (f(x) - g(x)) = \lim_{x\to a} f(x) - \lim_{x\to a} g(x)\)
• The limit of a constant times a function: \(\lim_{x\to a} (c \cdot f(x)) = c \cdot \lim_{x\to a} f(x)\)

When calculating more intricate limits, these properties make the process straightforward.

Numerator and Denominator Limits

When dealing with rational expressions, such as \(\frac{2f(x)-3g(x)}{f(x)+g(x)}\), it's vital to evaluate the limits of both the numerator and denominator separately. This approach allows you to break down the problem into manageable parts. Here’s how it applies:

• Numerator Limit: This is the limit of the expression in the numerator. In this exercise, \(2f(x) - 3g(x)\) is evaluated as 2 times the limit of \(f(x)\) minus 3 times the limit of \(g(x)\).
• Denominator Limit: This is the limit for the denominator. Here, \(f(x) + g(x)\) is simplified to the sum of their individual limits.

By calculating these separately, we find the specific limit values before proceeding to solve the main limit of the entire fraction.

Substitution in Limits

Once the limit properties are applied to simplify the expression, it's often necessary to use substitution. This step involves replacing the functions at the specified limit value directly.
In the given problem, by substituting the known limits, \(\lim_{x\to a} f(x) = 3 \) and \( \lim_{x\to a} g(x) = -1 \), into the simplified expressions:

• The numerator becomes \(2 \times 3 - 3 \times (-1)\).
• The denominator turns into \(3 + (-1)\).

This use of substitution ensures that the function's behavior at a specific point \(a\) is accurately described.

Limit Rules

Limit rules are essential guidelines that simplify the calculation of unknown limits using known ones. In combination with limit properties, these rules help ensure accuracy when solving limit problems. Some key limit rules are:

• The limit of a quotient: \(\lim_{x\to a} \frac{f(x)}{g(x)} = \frac{\lim_{x\to a} f(x)}{\lim_{x\to a} g(x)}\), provided that \(\lim_{x\to a} g(x) eq 0\).
• Product Rule: \(\lim_{x\to a} (f(x)\cdot g(x)) = \lim_{x\to a} f(x) \cdot \lim_{x\to a} g(x)\)
• L'Hôpital's Rule (when limits result in indeterminate forms like \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\)): \(\lim_{x\to a} \frac{f(x)}{g(x)} = \lim_{x\to a} \frac{f'(x)}{g'(x)}\). This rule is applicable under specific conditions.

By using these rules, especially the quotient rule, we can confidently determine that the overall limit for our function is \(4.5\). This demonstrates the utility of limit rules in achieving precise results.