Problem 32 - Thomas Calculus

Question

\(\infty / \infty\) Form Give an example of two differentiable functions \(f\) and \(g\) with \(\lim _{x \rightarrow \infty} f(x)=\lim _{x \rightarrow \infty} g(x)=\infty\) that satisfy the following. $$ \begin{array}{ll}{\text { a. } \lim _{x \rightarrow \infty} \frac{f(x)}{g(x)}=3} & {\text { b. } \lim _{x \rightarrow \infty} \frac{f(x)}{g(x)}=0} \\ {\text { c. } \lim _{x \rightarrow \infty} \frac{f(x)}{g(x)}=\infty}\end{array} $$

Step-by-Step Solution

Answer

1. \(f(x)=3x\), \(g(x)=x\); 2. \(f(x)=x\), \(g(x)=x^2\); 3. \(f(x)=x^2\), \(g(x)=x\).

Step 1: Solve part (a) of the exercise

We need two differentiable functions \(f(x)\) and \(g(x)\) such that as \(x\) approaches infinity, the ratio \(\frac{f(x)}{g(x)}\) is equal to 3. A simple example is to choose \(f(x) = 3x\) and \(g(x) = x\). As \(x\) approaches infinity, \(\lim_{x \to \infty} \frac{3x}{x} = \lim_{x \to \infty} 3 = 3\).

Step 2: Solve part (b) of the exercise

For this part, we need \(f(x)\) and \(g(x)\) where the limit of their ratio is 0 as \(x\) approaches infinity. Choose \(f(x) = x\) and \(g(x) = x^2\). Then \(\frac{f(x)}{g(x)} = \frac{x}{x^2} = \frac{1}{x}\). As \(x\) approaches infinity, \(\lim_{x \to \infty} \frac{1}{x} = 0\).

Step 3: Solve part (c) of the exercise

For the last part, select functions where the limit of their ratio is infinity as \(x\) approaches infinity. Let \(f(x) = x^2\) and \(g(x) = x\). Then \(\frac{f(x)}{g(x)} = \frac{x^2}{x} = x\). As \(x\) approaches infinity, \(\lim_{x \to \infty} x = \infty\).

Key Concepts

Differentiable Functions

Understanding differentiable functions is key to grasping the concept of limits involving the division of functions. A function is considered differentiable if it has a derivative at every point in its domain. This essentially means that the function is smooth and has no sharp corners or breaks. Differentiability implies continuity. This means if a function is not continuous at a point, it cannot be differentiable at that point, too.

• The derivative of a function tells us how the function's output changes as we change the input slightly.
• For example, if you have a function representing distance traveled over time, the derivative tells us the speed at every point in time.

Differentiable functions are crucial in calculus as they allow us to calculate limits, slopes, and rates of change efficiently. When working with these functions as they approach infinity, we can use their derivatives to predict behavior and even to find limits within specific conditions.

Ratio of Functions

The ratio of two functions, written as \(\frac{f(x)}{g(x)}\), can provide insights into their behavior as they grow large. This ratio helps determine how one function scales compared to another. Calculating the limit of this ratio often involves looking at the highest powers of \(x\) in both the numerator and denominator.

• If the highest power in the numerator matches that of the denominator, the coefficients determine the limit.
• If the numerator has a higher degree, the ratio will typically approach infinity.
• If the numerator has a lower degree, the ratio approaches zero.

Analyzing such ratios can show whether one function overtakes another or if they grow proportionately. Understanding the behavior of these ratios requires understanding algebraic manipulation and applying limits as \(x \to \infty\).

Infinity Limits

Infinity limits help us understand how functions behave as the input increases or decreases without bound. These limits are vital for understanding asymptotic behavior, which is crucial in mathematical modeling and real-world applications. The concept of an infinity limit doesn't mean the function "reaches" infinity, but rather describes the trend of its values.

• If \( \lim_{x \to \infty} f(x) = L \), it means as \(x\) becomes very large, \(f(x)\) approaches the value \(L\).
• If \( \lim_{x \to \infty} f(x) = \infty \), the function grows indefinitely without bound.
• If \( \lim_{x \to \infty} \frac{f(x)}{g(x)} = 0 \), it indicates that \(f(x)\) grows slower compared to \(g(x)\). For example, comparing \(x\) to \(x^2\), where \(x^2\) overtakes \(x\) as \(x\) grows.

Using these concepts, we can solve various problems that predict how functions behave at extreme values, allowing to understand more complex mathematical structures and their applications to physics or economics.