Problem 55 - Thomas Calculus

Question

Suppose \(\lim _{x \rightarrow b} f(x)=7\) and \(\lim _{x \rightarrow b} g(x)=-3 .\) Find \begin{equation} \quad \quad \quad\text { a. }\lim _{x \rightarrow b}(f(x)+g(x)) \quad \text { b. } \lim _{x \rightarrow b} f(x) \cdot g(x) \\\ \text { c. }\lim _{x \rightarrow b} 4 g(x) \quad \text { d. } \lim _{x \rightarrow b} f(x) / g(x) \end{equation}

Step-by-Step Solution

Answer

a. 4; b. -21; c. -12; d. -\frac{7}{3}.

Step 1: Understanding the Sum of Limits

To find \( \lim_{x \rightarrow b}(f(x) + g(x)) \), we use the property that the limit of a sum is the sum of the limits. Thus, \( \lim_{x \rightarrow b} (f(x) + g(x)) = \lim_{x \rightarrow b} f(x) + \lim_{x \rightarrow b} g(x) \). Substituting the given limits, we have \( 7 + (-3) \).

Step 2: Calculating the Sum of Limits

The sum \( 7 + (-3) = 4 \). Therefore, \( \lim_{x \rightarrow b} (f(x) + g(x)) = 4 \).

Step 3: Understanding the Product of Limits

To find \( \lim_{x \rightarrow b} f(x) \cdot g(x) \), we use the property that the limit of a product is the product of the limits. Thus, \( \lim_{x \rightarrow b} (f(x) \cdot g(x)) = \lim_{x \rightarrow b} f(x) \cdot \lim_{x \rightarrow b} g(x) \).

Step 4: Calculating the Product of Limits

Multiply the given limits: \( 7 \cdot (-3) = -21 \). Therefore, \( \lim_{x \rightarrow b} f(x) \cdot g(x) = -21 \).

Step 5: Understanding the Scalar Multiplication of Limits

To find \( \lim_{x \rightarrow b} 4g(x) \), we apply the property where a constant can be factored out of the limit. Thus, \( 4 \cdot \lim_{x \rightarrow b} g(x) = 4 \cdot (-3) \).

Step 6: Calculating the Scalar Multiplication

Perform the multiplication: \( 4 \cdot (-3) = -12 \). Therefore, \( \lim_{x \rightarrow b} 4g(x) = -12 \).

Step 7: Understanding the Quotient of Limits

To find \( \lim_{x \rightarrow b} \frac{f(x)}{g(x)} \), use the property that the limit of a quotient is the quotient of the limits, provided the denominator is not zero: \( \frac{\lim_{x \rightarrow b} f(x)}{\lim_{x \rightarrow b} g(x)} = \frac{7}{-3} \).

Step 8: Calculating the Quotient of Limits

Divide the limits: \( \frac{7}{-3} = -\frac{7}{3} \). Therefore, \( \lim_{x \rightarrow b} \frac{f(x)}{g(x)} = -\frac{7}{3} \).

Key Concepts

Limit laws

When studying calculus, limits are a fundamental concept that allow us to understand the behavior of functions as they approach certain points. Limit laws are a set of rules that simplify the process of calculating limits. These laws are grounded in the idea that limits respect operations much like numbers do.
Some of the common limit laws include:

• The sum rule, where the limit of a sum is the sum of the limits.
• The product rule, where the limit of a product is the product of the limits.
• The quotient rule, which holds as long as the limit of the denominator isn't zero.
• The constant multiple rule, which allows you to factor out a constant from the limit.

These principles make it easier to break down complex expressions into simpler, more manageable parts. If you're computing limits for a function that combines multiple simpler functions, these laws will help make your calculations more straightforward.

Sum of limits

The sum of limits is one of the simplest yet powerful rules in calculus. It states that the limit of a sum is equal to the sum of the limits. In mathematical terms, this is expressed as:
\[\lim_{x \to b} (f(x) + g(x)) = \lim_{x \to b} f(x) + \lim_{x \to b} g(x)\]In practice, if you know the individual limits of two functions as they approach a point, you can find the limit of their sum by simply adding these values together. In our example, we have \( \lim_{x \to b} f(x) = 7 \) and \( \lim_{x \to b} g(x) = -3 \). Thus, \( \lim_{x \to b} (f(x) + g(x)) = 7 + (-3) = 4 \).
This method greatly simplifies the process of finding limits for functions defined as sums, reducing complex calculations to simple arithmetic.

Product of limits

The product of limits is another key rule in the toolbox of limit laws. This rule asserts that the limit of a product is simply the product of the limits, allowing easy computation of the limit for products of functions. It is represented by:
\[\lim_{x \to b} (f(x) \cdot g(x)) = \lim_{x \to b} f(x) \cdot \lim_{x \to b} g(x)\]For instance, given \( \lim_{x \to b} f(x) = 7 \) and \( \lim_{x \to b} g(x) = -3 \), the limit of their product is computed by multiplying these limits: \( 7 \cdot (-3) = -21 \). Therefore, \( \lim_{x \to b} (f(x) \cdot g(x)) = -21 \).
This property makes calculations efficient by transforming complex function products into simple arithmetic operations, maintaining the functional relationship like multiplication does with numbers.

Quotient of limits

The quotient of limits provides a guideline for solving limits involving division. According to this rule, the limit of a quotient equals the quotient of the limits, with the condition that the denominator's limit is not zero. The formula is:
\[\lim_{x \to b} \frac{f(x)}{g(x)} = \frac{\lim_{x \to b} f(x)}{\lim_{x \to b} g(x)}\]In the given exercise, if \( \lim_{x \to b} f(x) = 7 \) and \( \lim_{x \to b} g(x) = -3 \), the quotient becomes \( \frac{7}{-3} = -\frac{7}{3} \).
It’s crucial to ensure that the denominator's limit is not zero, as division by zero is undefined. This rule thus enables the simplification of limits in divisions, transforming potentially complex expressions into straightforward calculations.