Problem 18 - 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. \(\lim _{x \rightarrow 1} \frac{f(x)}{h(x)}\)

Step-by-Step Solution

Answer

Question: Evaluate the limit of the function \(\frac{f(x)}{h(x)}\) as \(x\) approaches 1, given that \(\lim_{x \rightarrow 1} f(x) = 8\) and \(\lim_{x \rightarrow 1} h(x) = 2\). State the limit laws used in your calculations. Answer: The limit of the function \(\frac{f(x)}{h(x)}\) as \(x\) approaches 1 is 4. We used the quotient law in our calculations.

Step 1: Analyze the expression and apply limit laws

To compute the limit of the expression \(\frac{f(x)}{h(x)}\) as \(x\) approaches 1, we need to apply the following limit laws: 1. The quotient law which states that the limit of a quotient is equal to the quotient of the limits, provided the limit of the denominator is not equal to zero. In this case, we have \(\lim_{x \rightarrow 1} \frac{f(x)}{h(x)} = \frac{\lim_{x \rightarrow 1} f(x)}{\lim_{x \rightarrow 1} h(x)}\) , given that \(\lim_{x \rightarrow 1} h(x) \neq 0\).

Step 2: Substitute the limits

Now, substitute the given limits for each function: \(\lim_{x \rightarrow 1} f(x) = 8\) \(\lim_{x \rightarrow 1} h(x) = 2\)

Step 3: Calculate the limit

By applying the quotient law and using the given limits, we have: \(\lim_{x \rightarrow 1} \frac{f(x)}{h(x)} = \frac{\lim_{x \rightarrow 1} f(x)}{\lim_{x \rightarrow 1} h(x)}= \frac{8}{2}\)

Step 4: Simplify the result

Simplify the expression by dividing the numerator by the denominator: \(\frac{8}{2} = 4\) We have now calculated the limit of \(\frac{f(x)}{h(x)}\) as \(x\) approaches 1 and used the quotient law to justify our computation. The limit is 4.

Key Concepts

Quotient Law of Limits

When exploring the concept of limits in calculus, the Quotient Law of Limits emerges as a pivotal rule. It informs us that the limit of a ratio is the ratio of the limits, assuming of course the limit of the denominator isn't zero, since division by zero is undefined. This law is elegantly simple:
\[ \lim_{{x \to a}} \frac{{f(x)}}{{g(x)}} = \frac{{\lim_{{x \to a}} f(x)}}{{\lim_{{x \to a}} g(x)}} \]
Applying this to the given exercise simplifies the process of finding the limit of \( \frac{{f(x)}}{{h(x)}} \) when \( x \) approaches 1. This foundational principle reinforces that limits can be manipulated in a manner similar to ordinary numbers, providing they meet the necessary conditions.

Limit Computation

The art of computing limits is a fundamental skill in calculus, essential for understanding how functions behave as they approach a specific point. To compute a limit like \( \lim_{{x \to 1}} \frac{{f(x)}}{{h(x)}} \), we should strategically use limit laws alongside known limit values. In the provided exercise, the substitution of known limits after applying the quotient law manifested in a clear answer. It's essential to note, however, that accuracy in this process depends on previously established limit values of individual functions at the point of interest. With correct values on hand, and a methodical application of limit laws, the complexity of limit computation can be significantly reduced.

Calculus

Calculus is an extensive field of mathematics that studies change and motion; limits are at its core. Through differential and integral calculus, students learn to analyze functions, solve problems involving rates of change, and find areas under curves. In the context of limits, calculus teaches us how to predict function behavior as we approach a particular point, even when direct substitution isn't possible. For this type of mathematical exploration, understanding limit laws, including the Quotient Law, is crucial. They serve as tools to dissect and compute limits, providing a way to explore function behavior at points of discontinuity or where the function is not explicitly defined.