Problem 19 - Calculus Early Transcendentals

Question

Assume \(\lim _{x \rightarrow 1} f(x)=8, \lim _{x \rightarrow 1} g(x)=3,\) and \(\lim _{x \rightarrow 1} h(x)=2 .\) Compute the following limits and state the limit laws used to justify your computations. \(\lim _{x \rightarrow 1}(f(x)-g(x))\)

Step-by-Step Solution

Answer

Answer: The limit of the function (f(x)-g(x)) as x approaches 1 is 5.

Step 1: Use the limit laws

Since we want to find the limit of the difference between two functions f(x) and g(x), we can apply the difference rule for limits (which states that the limit of a difference is the difference of the limits): \(\lim _{x \rightarrow 1}(f(x)-g(x)) = \lim _{x \rightarrow 1} f(x) - \lim _{x \rightarrow 1} g(x)\)

Step 2: Apply the given limits

Next, we use the given limits of the functions f(x) and g(x): \(\lim _{x \rightarrow 1} f(x) = 8\) \(\lim _{x \rightarrow 1} g(x) = 3\)

Step 3: Substitute the limits

Now we substitute the limits into the original equation: \(\lim _{x \rightarrow 1}(f(x)-g(x)) = 8 - 3\)

Step 4: Compute the Result

Lastly, we compute the result: \(\lim _{x \rightarrow 1}(f(x)-g(x)) = 5\) In conclusion, \(\lim _{x \rightarrow 1}(f(x)-g(x)) = 5\). We have used the difference rule for limits to justify our computations.

Key Concepts

Limit Laws

Understanding limits is crucial in calculus as they help describe the behavior of a function as it approaches a specific point. Limit laws are fundamental rules that allow us to simplify and evaluate limits with efficiency. Some notable limit laws include:

• **Sum Rule**: This law states that the limit of a sum of functions equals the sum of their limits. For functions \(f(x)\) and \(g(x)\), it is expressed as \(\lim_{x \to a} [f(x) + g(x)] = \lim_{x \to a} f(x) + \lim_{x \to a} g(x)\).
• **Product Rule**: The limit of the product of functions is the product of their limits: \(\lim_{x \to a} [f(x) \cdot g(x)] = \lim_{x \to a} f(x) \cdot \lim_{x \to a} g(x)\).
• **Quotient Rule**: For the quotient of two functions, the limit is the quotient of their limits, provided the limit of the denominator is not zero: \(\lim_{x \to a} \frac{f(x)}{g(x)} = \frac{\lim_{x \to a} f(x)}{\lim_{x \to a} g(x)}\).
• **Constant Multiple Rule**: This rule holds that the limit of a constant times a function is the constant times the limit of the function: \(\lim_{x \to a} [c \cdot f(x)] = c \cdot \lim_{x \to a} f(x)\).

By using these rules, you can often find limits more quickly, avoiding more complex algebraic manipulation. These laws serve as building blocks for finding limits efficiently.

Difference Rule

The difference rule for limits specifically addresses the scenario where you need to find the limit of a difference between two functions. This rule is straightforward and highly useful, especially when dealing with straightforward subtraction of functions:The difference rule states that the limit of a difference is equal to the difference of the limits. Mathematically, for two functions \(f(x)\) and \(g(x)\), this can be expressed as:\[\lim_{x \to a} [f(x) - g(x)] = \lim_{x \to a} f(x) - \lim_{x \to a} g(x)\]This rule is just one part of the larger framework of limit laws and helps make complex limit problems more manageable.When working with the problem \(\lim_{x \to 1}(f(x) - g(x))\) given the individual limits, you can directly apply this rule once you know:

• \(\lim_{x \to 1} f(x) = 8\)
• \(\lim_{x \to 1} g(x) = 3\)

Using the difference rule, it becomes straightforward to compute:\[8 - 3 = 5\]This elegance showcases the efficiency and power of using limit laws for solving calculus problems.

Computing Limits

Computing limits can seem abstract at first, but it is actually about evaluating how a function behaves as it approaches a particular point. To compute a limit effectively and accurately, follow these general steps:- **Substitution**: Attempt direct substitution of the point into the function. If you get a defined result, that is the limit.- **Apply Limit Laws**: Use the established limit laws to break down more complex expressions into simpler parts. This is where rules like the sum, product, difference, or constant multiple can play a significant role.- **Algebraic Manipulation**: If direct substitution or limit laws are initially unsuccessful, consider algebraic techniques to simplify the expression, such as factoring, expanding, or simplifying fractions.- **Graphical Approach**: Sometimes, considering the graph of the function near the point can provide insightful intuition for understanding the behavior and confirming the limit.When solving our specific example, \(\lim_{x \to 1}(f(x) - g(x))\), we utilized the difference rule directly. Substitution using the given values simplified the task, ultimately showing the result as 5. Understanding these techniques is vital for working through the journey of limits in calculus, making complex evaluations more intuitive and straightforward.