82–89. Comparing growth rates Determine which of the two functions grows faster, or state that they have comparable growth rates.


eˣ and 3ˣ

Verified step by step guidance

1. Step 1: Understand the concept of growth rates. Growth rate refers to how quickly a function increases as its input becomes larger. In this problem, we are comparing the growth rates of two exponential functions: eˣ and 3ˣ.
2. Step 2: Consider the base of the exponential functions. The function eˣ has a base of e (approximately 2.718), while the function 3ˣ has a base of 3. Generally, for exponential functions, a larger base results in faster growth.
3. Step 3: Analyze the behavior of the functions as x approaches infinity. As x becomes very large, the function with the larger base will grow faster. Since 3 is greater than e, we expect 3ˣ to grow faster than eˣ.
4. Step 4: Use the concept of limits to compare the growth rates formally. Consider the limit of the ratio of the two functions as x approaches infinity: lim(x→∞) (eˣ / 3ˣ). If this limit approaches 0, it indicates that 3ˣ grows faster than eˣ.
5. Step 5: Calculate the limit using properties of exponents. Rewrite the ratio as (e/3)ˣ and evaluate the limit: lim(x→∞) (e/3)ˣ. Since e/3 is less than 1, the limit approaches 0, confirming that 3ˣ grows faster than eˣ.

Key Concepts

Exponential Functions

Exponential functions are mathematical expressions of the form f(x) = a^x, where 'a' is a constant base and 'x' is the exponent. These functions grow rapidly as 'x' increases, and their growth rate is determined by the base. In this case, e^x and 3^x are both exponential functions, with 'e' being approximately 2.718 and '3' being the base of the second function.

Growth Rate Comparison

To compare the growth rates of two functions, we often analyze their limits as x approaches infinity. If one function grows significantly faster than the other, we can conclude that it dominates in terms of growth. In this scenario, we will evaluate the limit of the ratio of e^x to 3^x as x approaches infinity to determine which function grows faster.

L'Hôpital's Rule

L'Hôpital's Rule is a method used to evaluate limits of indeterminate forms, such as 0/0 or ∞/∞. It states that if the limit of f(x)/g(x) results in an indeterminate form, we can take the derivative of the numerator and the denominator separately and then re-evaluate the limit. This rule can be applied to the functions e^x and 3^x to analyze their growth rates effectively.