Question

\(\begin{array}{llll}\text { Comparing } & \text { Functions } & \text { In Exercises } & \mathbf{4 5}-\mathbf{5 0}, & \text { use } & \text { L'Hôpital's }\end{array}\) Rule to determine the comparative rates of increase of the functions \(f(x)=x^{m}, \quad g(x)=e^{n x},\) and \(h(x)=(\ln x)^{n}\) where \(n>0, m>0,\) and \(x \rightarrow \infty\). \(\lim _{x \rightarrow \infty} \frac{x^{m}}{e^{n x}}\)

Short answer

The limit as \(x\) approaches infinity of the quotient of \(x^{m}\) over \(e^{n x}\) is 0.

Step-by-step solution

Derivate both functions

We first need to find the derivatives of both \(f(x) = x^{m}\) and \(g(x) = e^{n x}\). Using the power rule, the derivative of \(x^{m}\) is \(m \cdot x^{m-1}\). For \(e^{n x}\), the derivative is \(n \cdot e^{n x}\), thanks to the chain rule.

Apply L'Hôpital's Rule

Now we apply L'Hôpital's Rule. It states that \(\lim_{x \rightarrow a} \frac{f(x)}{g(x)} = \lim_{x \rightarrow a} \frac{f'(x)}{g'(x)}\) if \(f(a) = g(a) = 0\) or \(f(a) = g(a) = \infty\). Applying the rule to our case, we get \(\lim_{x \rightarrow \infty} \frac{m \cdot x^{m-1}}{n \cdot e^{n x}}\).

Analyze the new limit

In this case, as \(x \rightarrow \infty\), \(x^{m-1}\) also approaches infinity, but \(e^{n x}\) grows exponentially and faster, so \(x^{m-1}/e^{nx}\) is going to 0. Therefore, the limit is 0 times a finite number, which is also 0.

Key concepts

Calculus

Calculus is the branch of mathematics that deals with the study of rates of change and the accumulation of quantities. It's split into two main areas: differential calculus, concerning rates of change and slopes of curves; and integral calculus, focusing on the accumulation of quantities and the areas under and between curves. Understanding calculus is vital for analyzing and predicting change, which makes it a foundational tool in physics, engineering, economics, and various other fields.

In the context of our example exercise, calculus applies to finding the comparative rates of increase. For the functions under consideration, we engage differential calculus to explore how rapidly each function grows as we increase the value of the variable significantly.

Limits

The concept of a limit is a fundamental building block of calculus. Limits describe the behavior of functions as they approach a specific point or infinity. They allow us to define derivatives, integrals, and the continuity of functions.

In the given exercise, determining the limit of the ratio of two functions as the variable approaches infinity helps to derive comparative rates of increase. When taking the limit of a function, we are not always interested in its exact value at a point, but rather its trend or direction – which can be conceptualized as what value the function 'approaches' as the input gets very large or very small.

Derivatives

Derivatives are the core feature of differential calculus. They measure the rate at which a quantity changes. Mathematically, the derivative of a function at a point is the slope of the tangent line to the function's graph at that point. It gives us an exact rate of change at any given moment.

In the solution to our comparative rates problem, we take the derivatives of both functions to apply L'Hôpital's Rule. The derivative tells us how the functions behave as they change – crucial for understanding how one function grows in comparison to another as the variable expands towards infinity.

Rates of Increase

The rate of increase is a measure of how quickly a quantity expands over time. In mathematics, when we talk about functions, the rate of increase is indicated by the slope or steepness of the graph. Higher rates of increase are represented by steeper slopes, while lower rates are indicated by flatter slopes.

Our example exercise compares the rates of increase between polynomial and exponential functions. By using L'Hôpital's Rule and evaluating derivatives, we can determine that polynomial functions increase at a slower rate compared to exponential functions as the variable tends towards infinity.

Exponential Growth

Exponential growth occurs when a quantity increases at a consistent rate relative to its current value, resulting in the quantity growing faster and faster as it gets larger. This concept is often visualized on a graph by a curved line that starts off gently and then rapidly shoots upwards.

The function represented by e^{nx} in our problem is a classic example of exponential growth. As x increases, the value of e^{nx} grows at an increasingly rapid rate. In the world around us, exponential growth can be seen in populations, compound interest, and certain types of spreading phenomena like viruses.

Comparative Rates of Growth

Comparative rates of growth analyze how different functions increase relative to each other. By comparing the rates of growth, we can predict which function will eventually outgrow the other over time when their variables increase. This comparison is crucial in disciplines such as computer science, demographics, and financial modeling, where it's necessary to predict long-term trends.

In our textbook exercise, we're interested in comparing the growth rates of polynomial, exponential, and logarithmic functions. Using L'Hôpital's Rule, it becomes evident that exponential functions increase at a rate that eventually surpasses both polynomial and logarithmic functions as x moves toward infinity, leading to a deeper understanding of the long-term behavior of diverse mathematical models.