Problem 63 - A Graphical Approach to Precalculus with Limits

Question

Find two functions \(f(x)\) and \(g(x)\) with the given properties. $$\lim _{x \rightarrow x} f(x)=\infty, \lim _{x \rightarrow x} g(x)=\infty, \text { and } \lim _{x \rightarrow x_{0}}[f(x)-g(x)]=\infty$$

Step-by-Step Solution

Answer

Choose \(f(x) = \frac{1}{(x-x_0)^2}\) and \(g(x) = \frac{1}{x-x_0}\).

Step 1: Identify Functions

We need to find two functions, \(f(x)\) and \(g(x)\), such that both \(f(x)\) and \(g(x)\) approach infinity as \(x\) approaches \(x_0\). A typical choice is to use functions that diverge to infinity, such as rational functions with vertical asymptotes.

Step 2: Choose f(x)

Define \(f(x)\) as \(f(x) = \frac{1}{(x-x_0)^2}\). This function approaches infinity as \(x\) approaches \(x_0\) because the denominator approaches 0.

Step 3: Choose g(x)

Define \(g(x)\) as \(g(x) = \frac{1}{x-x_0}\). Similar to \(f(x)\), this function also approaches infinity as \(x\) approaches \(x_0\).

Step 4: Evaluate Limit Difference

Calculate the limit of \(f(x) - g(x)\) as \(x\) approaches \(x_0\): \(\lim_{x\rightarrow x_0}(f(x) - g(x)) = \lim_{x\rightarrow x_0} \left( \frac{1}{(x-x_0)^2} - \frac{1}{x-x_0} \right)\).

Step 5: Simplify Expression

Factorize the expression: \((x-x_0)^{-2} - (x-x_0)^{-1} = \frac{1 - (x-x_0)}{(x-x_0)^2}\). As \(x\) approaches \(x_0\), the numerator becomes a constant minus an expression tending to 0, which causes the entire fraction to diverge to \(\infty\).

Step 6: Verify Solution

Check if all conditions are met: \(\lim_{x \to x_0} f(x) = \infty\), \(\lim_{x \to x_0} g(x) = \infty\), and \(\lim_{x \to x_0} [f(x) - g(x)] = \infty\). All conditions hold true with the functions chosen.

Key Concepts

Infinity Limits

In calculus, infinity limits are explored to understand the behavior of functions as they approach a certain point where the output becomes infinitely large.
Such limits are indicative of situations where functions do not approach finite values.
For instance, the functions \( f(x) = \frac{1}{(x-x_0)^2} \) and \( g(x) = \frac{1}{x-x_0} \) are examples of functions that approach infinity as \( x \) gets closer to \( x_0 \).

• These functions do not settle on a particular number, but instead, their values grow without bound.
• Studying infinity limits helps us understand the circumstances under which functions become unbounded, providing insights into their behavior near vertical asymptotes.

As learners, grasping these limits enables a deeper comprehension of crucial mathematical concepts involving divergence.

Rational Functions

Rational functions are those that are expressed as the ratio of two polynomials, such as \( f(x) = \frac{1}{(x-x_0)^2} \).
They are foundational in studying calculus, especially while dealing with limits and asymptotic behaviors.
Their structure allows them to exhibit interesting properties, such as approaching infinity near vertical asymptotes.

• These functions can be used to model real-world phenomena where quantities can become very large or very small in comparison to others.
• Rational functions like \( \frac{1}{(x-x_0)} \) exhibit tendencies to approach infinite values as their denominators approach zero, creating significant changes in function values.

Understanding these attributes allows students to predict and analyze behavior in complex mathematical scenarios.

Asymptotic Behavior

Asymptotic behavior refers to the tendency of a function to approach a given line as the input gets larger or approaches a certain value, especially observable in rational functions.
In particular, vertical asymptotes occur when the function's value tends toward infinity as the input nears a specific point.
For instance, in functions like \( f(x) = \frac{1}{(x-x_0)^2} \), as \( x \) approaches \( x_0 \), the function's values escalate dramatically, reflecting a vertical asymptote.

• This behavior is crucial in predicting the graph's tendency and direction near defined boundary points.
• By studying asymptotes, students can forecast how functions react at boundaries, a vital skill when graphing or analyzing complex functions.

Gaining fluency in understanding asymptotic behavior prepares learners for advanced calculus and analytical tasks across various applications.