Problem 5 - APEX Calculus

Question

You are given the following information: (a) \(\lim _{x \rightarrow 1} f(x)=0\) (b) \(\lim _{x \rightarrow 1} g(x)=0\) (c) \(\lim _{x \rightarrow 1} f(x) / g(x)=2\) What can be said about the relative sizes of \(f(x)\) and \(g(x)\) as \(x\) approaches \(1 ?\)

Step-by-Step Solution

Answer

\(f(x)\) is about twice as large as \(g(x)\) near \(x = 1\).

Step 1: Identify the Nature of Limits

We are given three specific limits. First, \( \lim _{x \rightarrow 1} f(x) = 0 \) and \( \lim _{x \rightarrow 1} g(x) = 0 \) tell us that both functions approach zero as \( x \) approaches 1. The third limit, \( \lim _{x \rightarrow 1} \frac{f(x)}{g(x)} = 2 \), indicates the behavior of the quotient of the two functions as \( x \) approaches 1.

Step 2: Interpret the Limit of the Quotient

The statement \( \lim _{x \rightarrow 1} \frac{f(x)}{g(x)} = 2 \) suggests that the function \( f(x) \) is approaching \( 2g(x) \) as \( x \) approaches 1. This means that \( f(x) \) is approximately twice the size of \( g(x) \) near \( x = 1 \).

Step 3: Analyze Relative Sizes

Given that \( f(x) \) and \( g(x) \) both approach zero and their quotient approaches 2, we conclude that near \( x = 1 \), the function \( f(x) \) is behaving like a function that is twice as large as \( g(x) \). This is significant because it shows that even though they both go to zero, they do so at a rate where \( f(x) \) consistently remains about twice as large as \( g(x) \).

Key Concepts

Limit of a Function

The concept of a limit is fundamental in calculus. It helps us understand how a function behaves as it approaches a certain point. When we say \( \lim_{x \to 1} f(x) = 0 \), it means that as \( x \) gets closer to 1, the values of \( f(x) \) are getting closer to 0. This does not necessarily mean \( f(x) \) actually equals zero at \( x = 1 \), rather it's about the trend.

Limits can provide insights even when the function isn't well-defined at the point. This approach helps in understanding possible discontinuities or indeterminate forms in functions. By analyzing limits, mathematicians can make sense of behaviors that aren't immediately obvious just by looking at function values at specific points.

Quotient of Limits

The limit of a quotient, like \( \lim_{x \to 1} \frac{f(x)}{g(x)} = 2 \), gives a powerful insight into how two functions relate to each other near a particular point. In this case, both \( f(x) \) and \( g(x) \) approach zero as \( x \) approaches 1, yet their quotient still approaches 2. Here's why this is fascinating:

• The numerator \( f(x) \) and the denominator \( g(x) \) are both shrinking towards zero.
• Despite that, their rates of approach are such that the fraction \( \frac{f(x)}{g(x)} \) balances to converge to 2.

This implies \( f(x) \) is decreasing towards zero twice as fast as \( g(x) \) is. Understanding the quotient of limits becomes crucial when dealing with indeterminate forms like \( \frac{0}{0} \), helping to clarify "how" two functions compare near the limit point.

Behavior of Functions near a Point

The behavior of functions near a specific point can yield crucial insights, especially when both functions approach zero. As sniffed from the given information, \( f(x) \) and \( g(x) \) both trend towards zero as \( x \) nears 1. However, their interaction as they approach zero reveals the richer story.

• This relation \( \lim_{x \to 1} \frac{f(x)}{g(x)} = 2 \) indicates \( f(x) \) is effectively \( 2g(x) \) as they approach the limit.
• Thus, even though they both diminish to zero, \( f(x) \) stays consistently larger (twice the size) compared to \( g(x) \).
• The consistency of this proportion highlights a significant balance between \( f(x) \) and \( g(x) \) despite their eventual disappearance.

This behavior near the point opens deeper understandings in calculus, particularly in handling functions that tend to zero but maintain a steady proportion, showcasing a subtle yet important harmony of functions.