Problem 37 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Evaluate the limits. (a) \(\lim _{x \rightarrow \infty}-5\left(\frac{2}{3}\right)^{x}+7\) (b) \(\lim _{x \rightarrow-\infty}-5\left(\frac{2}{3}\right)^{x}+7\)

Step-by-Step Solution

Answer

The limit of the function as x approaches \(\infty\) is 7, and the limit of the function as x approaches \(-\infty\) is also 7.

Step 1: Evaluate the limit at positive infinity

When \(x\) approaches \(\infty\), \(\frac{2}{3}^x\) will approach 0 since \(\frac{2}{3}\) is less than 1. Thus, as \(x\) tends to \(\infty\), \(-5 * (\frac{2}{3})^x\) tends to 0. Therefore, \(\lim_{{x} \to \infty} -5\left(\frac{2}{3}\right)^{x}+7 = 0+7=7.\)

Step 2: Evaluate the limit at negative infinity

When \(x\) approaches \(-\infty\), \(\left(\frac{2}{3}\right)^x\) becomes \(\left(\frac{3}{2}\right)^{-x}\) changing the exponent from negative to positive. Now \(\left(\frac{3}{2}\right)^{-x}\) tends to 0 as \(x\) tends to \(\infty\), so -5 times this term also tends towards 0. Therefore, \(\lim_{{x} \to -\infty} -5\left(\frac{2}{3}\right)^{x}+7 = 0+7=7.\)

Key Concepts

Infinity

Infinity is a fascinating concept in mathematics. It's a term used to describe something that is unbounded or limitless. When we say that a number approaches infinity, we mean it grows without any constraint towards no particular, finite number. In the context of limits, infinity (\(\infty\)) can be reached as part of the behavior of a function as it progresses along the x-axis.

Keep in mind the following points when dealing with infinity:

• Conceptually, infinity isn't a real number but an idea used to help us understand behavior in mathematics.
• A function heading towards infinity can either go towards positive infinity (\(\infty\)) or negative infinity (\(-\infty\)), depending on its direction.
• Even though infinity isn't finite, we often talk about limits approaching infinity in the context of describing how functions behave.

Understanding infinity helps us predict how variables behave when they grow exceedingly large or small.

Limits

Limits are a fundamental concept in calculus, capturing the idea of approaching a particular value. When we talk about limits, we're looking at what value a function tends toward as the input (or variable) moves nearer to a certain point, which could be a number or infinity.

For example, consider the statement \( \lim_{{x} \to \infty} -5\left(\frac{2}{3}\right)^{x}+7 = 7 \). This tells us that as \( x \) gets very large, the function \( -5 \left( \frac{2}{3} \right)^{x} + 7 \) approaches 7. This behavior happens because \( \left(\frac{2}{3}\right)^{x} \) becomes smaller when raised to high powers as \( x \) moves towards infinity.

• Limits can help us explore behavior at "infinite points," providing insights into the patterns of functions as variables head towards unattainable values.
• When dealing with limits, especially involving infinity, recognize that the end result is typically the constant part of the function when other terms vanish.

Exponential Functions

Exponential functions are functions where a constant base is raised to a variable exponent. They are commonly represented as \( a^x\). In these equations, \( a \) is a constant, and \( x \) is the variable.

In the problem being discussed, we have an exponential expression \( \left(\frac{2}{3}\right)^x \). The base, \( \frac{2}{3} \), is less than one. This characteristic leads to distinct behavior:

• When \( x \) increases towards positive infinity, the value of \( \left(\frac{2}{3}\right)^x \) will drop towards zero. This is because multiplying numbers less than one by themselves leads to progressively smaller numbers.
• If \( x \) moves towards negative infinity, we are essentially flipping the exponent to \(-x\), which can be rewritten with a reciprocal base like \( \left(\frac{3}{2}\right)^{-x} \), and as \( x \) progresses, this also goes to zero.

Exponential functions can have deep implications for growth and decay, depending on if the base is greater than, less than, or equal to 1.