Problem 65 - Calculus: Graphical, Numerical, Algebraic

Question

Exploring Properties of Limits Find the limits of \(f, g,\) and \(f g\) as \(x \rightarrow c .\) (a) $$f(x)=\frac{1}{x}, \quad g(x)=x, \quad c=0$$ (b) $$f(x)=-\frac{2}{x^{3}}, \quad g(x)=4 x^{3}, \quad c=0$$

Step-by-Step Solution

Answer

\((a) \lim_{{x \to 0}} f(x) = \infty, \lim_{{x \to 0}} g(x) = 0, \lim_{{x \to 0}} fg(x) = undefined;\) \n \((b) \lim_{{x \to 0}} f(x) = - \infty, \lim_{{x \to 0}} g(x) = 0, \lim_{{x \to 0}} fg(x) = undefined\)

Step 1: Compute the limits for (a)

For \(f(x) = \frac{1}{x}\), as \(x\) approaches \(0\), \(f(x)\) approaches infinity: \n \[\lim_{{x \to 0}} f(x) = \lim_{{x \to 0}} \frac{1}{x} = \infty\]. For \(g(x) = x\), as \(x\) approaches \(0\), \(g(x)\) approaches \(0\): \[\lim_{{x \to 0}} g(x) = \lim_{{x \to 0}} x = 0\]. To find the limit of the product \(fg\), apply the limit product rule: \[\lim_{{x \to 0}} fg(x) = \lim_{{x \to 0}} f(x)g(x) = \lim_{{x \to 0}} f(x) \cdot \lim_{{x \to 0}} g(x) = \infty \cdot 0 = undefined\].

Step 2: Compute the limits for (b)

For \(f(x) = -\frac{2}{x^3}\), as \(x\) approaches \(0\), \(f(x)\) approaches negative infinity: \[\lim_{{x \to 0}} f(x) = \lim_{{x \to 0}} -\frac{2}{x^3} = - \infty\]. For \(g(x) = 4x^3\), as \(x\) approaches \(0\), \(g(x)\) approaches \(0\): \[\lim_{{x \to 0}} g(x) = \lim_{{x \to 0}} 4x^3 = 0\]. To find the limit of the product \(fg\), apply the limit product rule: \[\lim_{{x \to 0}} fg(x) = \lim_{{x \to 0}} f(x)g(x) = \lim_{{x \to 0}} f(x) \cdot \lim_{{x \to 0}} g(x) = - \infty \cdot 0 = undefined\].

Key Concepts

Properties of Limits

When we study calculus limits, we often rely on a set of properties that make calculations more straightforward. These properties give us rules to follow when we want to find the limit of certain functions as they approach a specific point.

One of the key properties is that if the limits of two functions exist as they approach a point, then the limit of their sum is just the sum of their limits. Similarly, the limit of a difference of functions is the difference of their limits:

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

Another important property is the limit of a product:

• Product: The limit of a product is the product of the limits, but this only holds if both limits exist.\[\lim_{{x \to c}} [f(x) \cdot g(x)] = \lim_{{x \to c}} f(x) \cdot \lim_{{x \to c}} g(x)\]

These properties ease the process of finding limits and help simplify complex limit calculations.

Indeterminate Forms

Indeterminate forms can be tricky because they appear when evaluating limits that seem impossible to determine directly. The most common indeterminate forms are \(\frac{0}{0}\) and \({0} \cdot \infty\).

In our exercise, we encountered the indeterminate form \({\infty} \cdot {0}\). When the limit of the product of two functions becomes \(\infty \cdot 0\), we cannot straightforwardly determine the limit. This is due to the contradiction of multiplying something infinitely large by zero.

To resolve these situations, mathematicians use different techniques depending on the context:

• Algebraic manipulation: Simplifying the expression can sometimes clarify the limit.
• Using L'Hôpital's Rule: Particularly useful for \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\) forms.
• Substitution: Replacing part of the expression with another function to make limits solvable.

Spotting indeterminate forms quickly can help identify when you need to use special techniques to find accurate limits.

Product of Limits

The product of limits is a fundamental concept in calculus that tells us how to find the limit of a product of two functions. At first glance, this may seem straightforward: just multiply the limits of the individual functions.

However, like we saw in the earlier example, this can become complicated when one of the limits involves indeterminate forms.

The principle of the product of limits states:

• If \lim_{{x \to c}} f(x)\ and \lim_{{x \to c}} g(x)\ exist, then the limit of their product is simply:\[\lim_{{x \to c}} [f(x) \cdot g(x)] = \lim_{{x \to c}} f(x) \cdot \lim_{{x \to c}} g(x)\]

However, if either limit is not well-defined or approaches zero or infinity, like in our given exercise problem, we face indeterminate forms like \(\infty \cdot 0\).

In such cases, the product limit doesn't apply directly, and extra care is required to evaluate it accurately. Often, revisiting the strategy used to calculate the individual limits helps determine what such a product implies. Exploring these situations extends our capacity to handle complex calculus problems effectively. Understanding these concepts can empower us to better predict the behavior of functions beyond simple cases.