Problem 8 - 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

Answer: The limit of \(\frac{f(x)}{h(x)}\) as x approaches 1 is 4.

Step 1: Recall the limit properties

We need to recall the limit properties that allow us to compute the limit of \(\frac{f(x)}{h(x)}\). The specific property we will use is the Quotient rule, which states that if \(\lim _{x \rightarrow a} f(x) = L\) and \(\lim _{x \rightarrow a} g(x) = M\), then \(\lim _{x \rightarrow a} \frac{f(x)}{g(x)} = \frac{L}{M}\), provided that \(M \neq 0\).

Step 2: Apply the Quotient rule to \(\frac{f(x)}{h(x)}\)

We are given that \(\lim _{x \rightarrow 1} f(x) = 8\), \(\lim _{x \rightarrow 1} g(x) = 3\), and \(\lim _{x \rightarrow 1} h(x) = 2\). Applying the Quotient rule, we have: $$\lim _{x \rightarrow 1} \frac{f(x)}{h(x)} = \frac{\lim _{x \rightarrow 1} f(x)}{\lim _{x \rightarrow 1} h(x)}$$

Step 3: Substitute the limits and compute the result

Substituting the given limits, we get: $$\lim _{x \rightarrow 1} \frac{f(x)}{h(x)} = \frac{8}{2}$$ Now, simply divide the numbers: $$\lim _{x \rightarrow 1} \frac{f(x)}{h(x)} = 4$$ So, \(\lim _{x \rightarrow 1} \frac{f(x)}{h(x)}=4\). In this solution, we used the Quotient rule to justify our computation.

Key Concepts

Quotient Rule

Understanding the Quotient Rule is essential when dealing with the division of functions in limits. In calculus, when you have two functions divided by each other, and you want to find the limit of this quotient as the input approaches a certain value, the Quotient Rule makes this task manageable.

The rule simply states that if you have limits \( \lim _{x \rightarrow a} f(x) = L \) and \( \lim _{x \rightarrow a} g(x) = M \), then the limit of the quotient of these functions as \( x \) approaches \( a \) is \( \frac{L}{M} \) as long as \( M eq 0 \). The rationale behind this is that the limit of a division is equal to the division of the limits, provided the denominator isn't approaching zero.

For instance, in the given exercise, to find \( \lim _{x \rightarrow 1} \frac{f(x)}{h(x)} \), we know that \( \lim _{x \rightarrow 1} f(x) = 8 \) and \( \lim _{x \rightarrow 1} h(x) = 2 \). Applying the Quotient Rule, we can directly compute the limit as \( 4 \) without actually substituting \( x = 1 \) into \( f(x) \) and \( h(x) \) separately. It's a powerful shortcut that saves time and simplifies the process of finding limits for rational functions.

Limits in Calculus

Limits are a foundational concept in calculus that deal with approaching a specific value. When we say \( \lim _{x \rightarrow a} f(x) = L \), we mean that as \( x \) gets closer and closer to \( a \), the function \( f(x) \) approaches the value \( L \). Limits allow us to define many key concepts, such as continuity, derivatives, and integrals, making them exceedingly important.

What makes limits intriguing is their ability to analyze the behavior of functions at points where they may not be explicitly defined, such as at discontinuities or at points of infinity. They give us a glimpse into the 'trend' of function values as we get infinitesimally close to a certain point. This predictive power is why limits are so crucial to understanding change and rates of change in calculus.

Limit Properties

Limit properties are rules that make calculating complex limits more manageable. These properties include the Sum Rule, Product Rule, Quotient Rule, and others. They are based on the idea that limits can be manipulated in many of the same ways that regular numbers can. For example:

• The Sum Rule states that the limit of a sum is the sum of the limits.
• The Product Rule tells us that the limit of a product is the product of the limits.
• The Quotient Rule (as used in our example) assures that the limit of a quotient is the quotient of the limits, given the denominator isn't zero.
• There's also a Power Rule, stating that the limit of a function raised to a power is the limit of the function itself raised to that power.

When used correctly, these properties can dramatically simplify the process of finding limits. Furthermore, they give mathematicians and students alike the ability to break down complex functions into simpler parts, making it easier to see how a function behaves as the input approaches a particular value.