Problem 40 - Calculus Early Transcendentals: Pearson New International Edition

Question

Find examples to show that if (a) \(\lim _{x \rightarrow c}[f(x)+g(x)]\) exists, this does not imply that either \(\lim _{x \rightarrow c} f(x)\) or \(\lim _{x \rightarrow c} g(x)\) exists; (b) \(\lim _{x \rightarrow c}[f(x) \cdot g(x)]\) exists, this does not imply that either \(\lim _{x \rightarrow c} f(x)\) or \(\lim _{x \rightarrow c} g(x)\) exists.

Step-by-Step Solution

Answer

Examples: (a) \( f(x)=x \), \( g(x)=-x \); (b) \( f(x)=\sin\left(\frac{1}{x}\right) \), \( g(x)=\frac{1}{x} \).

Step 1: Understanding the Task

We need to find examples for two conditions: (a) where the limit of a sum of two functions exists but the individual limits do not, and (b) where the limit of a product of two functions exists but the individual limits do not.

Step 2: Example for Part (a) - Limit of Sum Exists

For (a), consider \( f(x) = x \) and \( g(x) = -x \). While \(f(x)\) and \(g(x)\) individually diverge as \(x\) approaches infinity, their sum \( f(x) + g(x) = x + (-x) = 0 \) clearly has a limit of 0 as \(x\) approaches infinity.

Step 3: Verify Limits for Part (a)

Check if \( \lim_{x \to \infty} f(x) \) and \( \lim_{x \to \infty} g(x) \) exist. We find that both \( \lim_{x \to \infty} f(x) = +\infty \) and \( \lim_{x \to \infty} g(x) = -\infty \) do not exist in the real number sense, but \( \lim_{x \to \infty} [f(x) + g(x)] = 0 \) exists.

Step 4: Example for Part (b) - Limit of Product Exists

For (b), consider \( f(x) = \sin\left(\frac{1}{x}\right) \) and \( g(x) = \frac{1}{x} \). As \(x\) approaches 0, \(g(x)\) approaches infinity, causing \(f(x)\) to oscillate. Their product \( f(x) \cdot g(x) = \sin\left(\frac{1}{x}\right) \cdot \frac{1}{x} \) approaches 0, since the oscillation of the sine function does not affect the product tending towards zero.

Step 5: Verify Limits for Part (b)

Analyze \( \lim_{x \to 0} f(x) \) and \( \lim_{x \to 0} g(x) \). Neither \( \lim_{x \to 0} \sin\left(\frac{1}{x}\right) \) exists (since it oscillates) nor does \( \lim_{x \to 0} \frac{1}{x} \) exist (approaches infinity), but \( \lim_{x \to 0} \left[ \sin\left(\frac{1}{x}\right) \cdot \frac{1}{x} \right] = 0 \).

Key Concepts

Limit of a Function

The limit of a function explores the behavior of the function as the input approaches a specific value. It allows us to understand the function's tendency without necessarily evaluating the function at that point. This concept is key in calculus and is foundational for continuity and differentiability.
When we say that a function \( f(x) \) has a limit \( L \) as \( x \) approaches \( c \), denoted as \( \lim_{x \to c} f(x) = L \), it means that as \( x \) gets closer to \( c \), \( f(x) \) approaches \( L \).
For example, consider the function \( f(x) = 3x + 2 \). As \( x \) approaches 2, the function \( f(x) \) approaches 8, so \( \lim_{x \to 2} (3x + 2) = 8 \).
Understanding limits helps in analyzing scenarios where functions may behave unpredictably or diverge at specific points.

Sum of Functions Limit

The sum of functions limit states that the limit of the sum of two functions is equal to the sum of their individual limits, provided these limits exist. This is given by: \( \lim_{x \to c} [f(x) + g(x)] = \lim_{x \to c} f(x) + \lim_{x \to c} g(x) \). This principle makes calculating limits simpler.
However, this rule requires the individual limits to exist. In our exercise example, we saw that if \( f(x) = x \) and \( g(x) = -x \), both individually diverge as \( x \to \infty \), but their sum \( f(x) + g(x) = 0 \) converges to 0. Here, even if the sum has a defined limit, the individual functions do not.
This demonstrates that sometimes, even if the sum of the functions behaves nicely, the individual functions may not be well-behaved.

Product of Functions Limit

The limit of a product of functions is another important rule in calculus, where \( \lim_{x \to c} [f(x) \cdot g(x)] = \lim_{x \to c} f(x) \cdot \lim_{x \to c} g(x) \), provided both limits individually exist. This is useful in determining the overall behavior of the functions.
In the exercise provided, with \( f(x) = \sin\left(\frac{1}{x}\right) \) and \( g(x) = \frac{1}{x} \), although neither \( \lim_{x \to 0} f(x) \) nor \( \lim_{x \to 0} g(x) \) exist, the product \( f(x) \cdot g(x) \) approaches 0. This happens because \( \sin(\frac{1}{x}) \)'s oscillations get smaller and smaller as \( x \) approaches 0, leading the product to converge.
Thus, the product can have a finite limit even if individual functions do not.

Diverging Functions

Diverging functions are those that do not approach a finite limit as the input grows infinitely large or decreases to zero. Such functions either increase or decrease without bound, or oscillate infinitely as they approach specific points.
For instance, in our example, the function \( f(x) = x \) diverges as \( x \to \infty \) because it increases indefinitely. Similarly, \( g(x) = -x \) also diverges negatively. Despite this, their sum yields a defined limit at certain points.
In another scenario, \( \sin\left(\frac{1}{x}\right) \) oscillates as \( x \to 0 \) and \( \frac{1}{x} \) diverges by becoming infinitely large. Individually, they do not have limits at zero, but together in a product, they converge to zero.
Understanding and identifying diverging behavior of functions is crucial to mastering limits and the broader topics in calculus.