Question

Use limit methods to determine which of the two given functions grows faster or state that they have comparable growth rates. $$e^{x^{2}} ; e^{10 x}$$

Short answer

Answer: The function \(e^{x^2}\) grows faster than the function \(e^{10x}\) as x goes to infinity.

Step-by-step solution

Rewrite the functions in terms of the exponent

We need to rewrite the functions in terms of the exponent to analyze the growth rates more easily. The given functions are \(e^{x^{2}}\) and \(e^{10x}\). We can rewrite them as: $$f(x) = e^{x^{2}}$$ $$g(x) = e^{10x}$$

Find the limit of the ratio of the functions as x goes to infinity

To determine which function grows faster or if they have comparable growth rates, we will find the limit of the ratio of the functions as \(x\) goes to infinity: $$\lim_{x \to \infty} \frac{f(x)}{g(x)} = \lim_{x \to \infty} \frac{e^{x^{2}}}{e^{10x}}$$

Simplify the limit

Using the properties of exponents, we can simplify the limit expression: $$\lim_{x \to \infty} \frac{e^{x^{2}}}{e^{10x}} = \lim_{x \to \infty} e^{x^{2} - 10x}$$

Determine if the limit exists as x goes to infinity

Now we will determine if the limit exists as \(x\) goes to infinity: $$\lim_{x \to \infty} e^{x^{2} - 10x}$$ As \(x\) approaches infinity, \(x^2\) will grow significantly faster than \(10x\). Therefore, the exponent will also increase without bound.

Evaluate the limit

Since the exponent is increasing without bound as \(x\) approaches infinity, the limit will also increase without bound: $$\lim_{x \to \infty} e^{x^{2} - 10x} = \infty$$

Determine which function grows faster

The limit of the ratio of the functions is infinity, which means that the function \(e^{x^{2}}\) grows faster than the function \(e^{10x}\) as \(x\) goes to infinity.

Key concepts

Limit Methods

Limit methods are essential tools in calculus to compare the growth rates of functions. They help in determining how fast a function grows as the variable approaches a certain value, often infinity. In this context, we calculate the limit of the ratio of two functions. By doing so, we assess which function overtakes the other as the input variable gets exceedingly large.
Suppose we have two functions, say \( f(x) \) and \( g(x) \). To compare their growth rates as \( x \) approaches infinity, we examine the limit:

• \( \lim_{x \to \infty} \frac{f(x)}{g(x)} \)

This ratio tells us several possibilities:

• If the limit equals zero, \( g(x) \) grows faster.
• If it's infinity, \( f(x) \) grows faster.
• If the limit is a positive number, both have comparable growth rates.

In our case, the limit \( \lim_{x \to \infty} \frac{e^{x^2}}{e^{10x}} = \lim_{x \to \infty} e^{x^2 - 10x} \) goes to infinity, indicating that \(e^{x^2}\) grows faster.

Exponential Growth

Exponential growth is a powerful concept where quantities increase more rapidly over time. The term is often used to describe how populations or investments grow, but it applies to mathematical functions, too. Exponential functions, like \( e^x \), feature a constant rate of growth that significantly compounds the output as the input grows larger.
In our problem, we have two exponential functions: \( e^{x^2} \) and \( e^{10x} \). The bases are the same, but the exponents differ, which impacts their growth rates.

• For \( e^{x^2} \), the exponent is quadratic, \( x^2 \), meaning it grows parabolically more than linearly.
• For \( e^{10x} \), the exponent is linear, \( 10x \), representing straightforward linear expansion.

Quadratic growth in the exponent leads to much faster growth compared to a linear increase, which is why \( e^{x^2} \) outstrips \( e^{10x} \) as \( x \) becomes very large.

Comparable Growth Rates

When evaluating functions, determining comparable growth rates can offer insight into how similar functions behave over time. Two functions have comparable growth rates if their ratio approaches a constant (other than zero or infinity) as \( x \) tends to infinity. This implies neither function greatly outpaces the other; they grow at similar speeds.
In our exercise, however, the two functions do not share comparable growth rates. The limit method we used showed that the term \( e^{x^2 - 10x} \) approaches infinity. This tells us that \( e^{x^2} \) grows much faster than \( e^{10x} \), indicating non-comparable growth rates for large \( x \).
Understanding comparable growth rates is useful in many fields such as computing long-term behavior of algorithms or systems in applied mathematics. It helps simplify problems where exact functions need not be distinguished but their overall growth trends are of interest.