Problem 44 - Calculus: Early Transcendentals

Question

Suppose that \( \displaystyle \lim_{x \to a} f(x) = \infty \) and \( \displaystyle \lim_{x \to a} g(x) = c \), where \( c \) is a real number. Prove each statement. (a) \( \displaystyle \lim_{x \to a} [ f(x) + g(x) ] = \infty \) (b) \( \displaystyle \lim_{x \to a} [ f(x) g(x) ] = \infty \) if \( c > 0 \) (c) \( \displaystyle \lim_{x \to a} [ f(x) g(x) ] = -\infty \) if \( c < 0 \)

Step-by-Step Solution

Answer

(a) \( \infty \), (b) \( \infty \) if \( c > 0 \), (c) \(-\infty\) if \( c < 0 \).

Step 1: Understand Given Limits

We are given that \( \lim_{x \to a} f(x) = \infty \) and \( \lim_{x \to a} g(x) = c \). This means as \( x \) approaches \( a \), \( f(x) \) gets arbitrarily large (increases without bound), while \( g(x) \) approaches a real number \( c \).

Step 2: Prove Limitation of Sum Function

(a) To prove \( \lim_{x \to a} [ f(x) + g(x) ] = \infty \), note that as \( x \to a \), \( f(x) \) becomes very large. Because \( g(x) \) approaches a real number \( c \), adding \( g(x) \) (which stabilizes) still results in \( f(x) + g(x) \) remaining large, driving the sum to \( \infty \). Thus, \( \lim_{x \to a} [ f(x) + g(x) ] = \infty \).

Step 3: Prove Limitation of Product Function (Case c > 0)

(b) For \( \lim_{x \to a} [ f(x) g(x) ] = \infty \) given \( c > 0 \), note that as \( x \to a \), \( g(x) \to c \). Therefore, \( f(x)g(x) \approx f(x)c \). Since \( f(x) \to \infty \) and \( c > 0 \), it follows that \( f(x)g(x) \to \infty \). Thus, \( \lim_{x \to a} [ f(x) g(x) ] = \infty \).

Step 4: Prove Limitation of Product Function (Case c < 0)

(c) Conversely, if \( c < 0 \), \( g(x) \to c \), making \( f(x)g(x) \approx f(x)c \). With \( f(x) \to \infty \) and \( c < 0 \), \( f(x)g(x) \to -\infty \) due to the negative multiplier \( c \). Therefore, \( \lim_{x \to a} [ f(x) g(x) ] = -\infty \).

Key Concepts

Infinity

Infinity is a fascinating concept in mathematics that represents an unbounded quantity. When we say a function, like \( f(x) \), has a limit of infinity as \( x \) approaches a value \( a \), we are expressing that \( f(x) \) grows larger and larger without ever settling on a finite number. In simpler terms, the values of the function keep increasing towards no upper bound, becoming indefinitely large.

Infinity is not a real number; it's more of an idea or a notion representing endlessness. It is important because it helps us understand concepts in calculus, such as limits and behaviors of functions, particularly in terms of "growing without bounds" as they approach specific points or infinity itself. In the context of limits, \( \lim_{x \to a} f(x) = \infty \) signifies that the function's value cannot be pinned down to any real number, highlighting its growth towards vastness.

Real Numbers

Real numbers are the set of numbers that include all rational and irrational numbers. This set encompasses everything from integers, fractions, and decimals to numbers that can’t be exactly expressed in fraction form, like \( \sqrt{2} \) or \( \pi \).

• Rational numbers: Can be expressed as a fraction or quotient of two integers, such as \( \frac{1}{2} \), \( 3.5 \), or -6.
• Irrational numbers: Cannot be expressed as a simple fraction and have non-repeating, non-terminating decimal expansions, such as \( \pi \) and the square root of any non-perfect square.

Real numbers are essential in calculus because they provide a comprehensible system for analyzing the behavior of functions at various points.

When we say \( \lim_{x \to a} g(x) = c \) where \( c \) is a real number, it implies that as \( x \) approaches \( a \), the function \( g(x) \) settles around this real number \( c \), unlike \( f(x) \) which flies off to infinity. This distinction is crucial when determining limits and understanding how functions operate near a point.

Continuous Functions

A continuous function is one where small changes in the input lead to small changes in the output. There are no sudden jumps, breaks, or holes in the graph of a continuous function. This means you can draw the graph of a continuous function without lifting your pencil off the paper.

For a function \( f(x) \) to be continuous at a point \( a \), three conditions must be satisfied:

• The function \( f(x) \) is defined at the point \( a \).
• \( \lim_{x \to a} f(x) \) exists.
• The limit \( \lim_{x \to a} f(x) \) equals the function's value at that point, \( f(a) \).

Continuous functions are important because they guarantee predictable behavior of functions near points of interest. In our example, as one function approaches a real number and another towards infinity, understanding continuity aids in predicting their behavior when combined. If a piece is part of a larger puzzle and behaves controllably, like \( g(x) \), it helps maintain perspective even if another piece, like \( f(x) \), behaves erratically and isn't anchored like \( g(x) \).

Function Behavior Near a Point

Understanding the behavior of functions near a point is a fundamental aspect of calculus. This involves analyzing how the function behaves as the input values approach a particular point, even if the functions themselves don't provide outputs for all these inputs.

In limits, the focus is on understanding the tendencies of functions as they get closer and closer to a specific point. We use limits to explore what value a function approaches as the input nears a given value.

• For \( f(x) \), approaching infinity implies boundless growth near the point \( x = a \).
• For \( g(x) \), approaching a real number \( c \) implies a stable, predictable value close to \( x = a \).

In our scenario, we apply this understanding of behavior to determine the outcomes of combined functions: how their sum or product behaves as our input gets arbitrarily close to \( a \). Each function's behavior near \( a \) helps set expectations about their interactions, which is insightful in proving statements like those given in the example.