Problem 50 - Calculus with Concepts in Calculus

Question

Suppose \(\lim _{x \rightarrow-\infty} f(x) / g(x)=1\) and \(\lim _{x \rightarrow-\infty} g(x)=\infty\) Show that \(\lim _{x \rightarrow-\infty} f(x)=\infty\)

Step-by-Step Solution

Answer

The limit of \(f(x)\) as \(x\) approaches \(-\infty\) is \(\infty\).

Step 1: Understanding the Given Limits

We are given that \(\lim _{x \rightarrow-\infty} \frac{f(x)}{g(x)}=1\). This means as \(x\) approaches negative infinity, the function \(\frac{f(x)}{g(x)}\) approaches 1. We are also given \(\lim _{x\rightarrow-\infty} g(x)=\infty\), indicating that as \(x\) becomes very large negatively, \(g(x)\) grows infinitely large.

Step 2: Rearranging the Limit Expression

From the limit \(\lim _{x \rightarrow-\infty} \frac{f(x)}{g(x)}=1\), we can deduce that \(f(x)\) is behaving similarly to \(g(x)\) for \(x\) approaching negative infinity. For the ratio to equal 1, both \(f(x)\) and \(g(x)\) must approach infinity, since \(g(x)\) approaches infinity.

Step 3: Using Limit Laws

Because \(\lim _{x \rightarrow-\infty} \frac{f(x)}{g(x)} = 1\), it implies \[ \lim _{x \rightarrow-\infty} f(x) = \lim _{x \rightarrow-\infty} 1 \cdot g(x) = \infty.\]This means that \(f(x)\) grows at a rate forced by \(g(x)\) to also approach infinity.

Key Concepts

Calculus

Calculus is fundamentally the mathematical study of change and motion. It provides tools to deal with quantities that are constantly changing, which are typically found in functions. Calculus explores two main concepts: differentiation and integration. Differentiation focuses on finding the rate at which a quantity changes, whereas integration involves finding the total accumulation of a quantity over an interval.
To tackle problems involving limits and infinite behavior in calculus, it is crucial to use a systematic approach, analyzing how functions behave as variables move towards infinity or negative infinity. Understanding these concepts is key to solving the exercise that deals with limits as they approach infinity.

Limit Laws

The limit laws are a set of rules in calculus that allow us to solve limit problems more effectively. These laws help simplify the process of finding limits for complex functions by breaking them down into more manageable parts. Here are some fundamental limit laws:

• Sum Law: \( \lim_{x \to c} [f(x) + g(x)] = \lim_{x \to c} f(x) + \lim_{x \to c} g(x) \)


• Product Law: \( \lim_{x \to c} [f(x) \cdot g(x)] = \lim_{x \to c} f(x) \cdot \lim_{x \to c} g(x) \)


• Quotient Law: \( \lim_{x \to c} \frac{f(x)}{g(x)} = \frac{\lim_{x \to c} f(x)}{\lim_{x \to c} g(x)} \), where \( \lim_{x \to c} g(x) eq 0 \).

The exercise uses these laws to conclude that if the limit of the quotient of two functions is 1 and the limit of the denominator is infinity, the numerator must also approach infinity.

Infinite Limits

Infinite limits refer to the behavior of functions as they approach infinity or negative infinity. When dealing with infinite limits, we are interested in how a function behaves as the input grows larger or smaller towards infinity or negative infinity.
An essential part of understanding infinite limits is recognizing how functions approach infinity. For example, if a function outputs a larger and larger value as the inputs go towards infinity, we say the limit of the function is infinity. In the exercise problem, the concept of infinite limits helps us recognize that both the numerator and denominator approach infinity, culminating in an infinite limit for the function's behavior.

Functions Behavior

The behavior of functions is closely examined in calculus, particularly how they change as the input moves towards certain values like infinity. To understand this, one must consider the properties of the functions involved. In the given exercise,
it is determined that the function \( f(x) \) behaves similarly to \( g(x) \) as \( x \to -\infty \). This suggests that \( f(x) \) grows without bound as \( g(x) \) does. The exercise emphasizes that for \( \lim_{x \to -\infty} \frac{f(x)}{g(x)} = 1 \) to hold true, \( f(x) \) must closely track \( g(x) \), especially as both functions move towards infinity.
Understanding functions' behaviors enables one to predict and explain their growth trends and limits, which are key to solving complex calculus problems like those involving infinite limits.