Problem 39 - Calculus for Biology and Medicine

Question

In Problems 29-48, find the limits. $$ \lim _{x \rightarrow 3} e^{x^{2}-9} $$

Step-by-Step Solution

Answer

The limit is 1.

Step 1: Understand the Limit Expression

We are asked to find the value of the expression as \(x\) approaches 3. The expression given is \(e^{x^2-9}\). This involves an exponential function with the exponent \(x^2 - 9\).

Step 2: Simplify the Exponent

Substitute \(x = 3\) into the exponent \(x^2 - 9\): \(3^2 - 9 = 9 - 9 = 0\). This simplifies the exponent of the exponential expression to 0.

Step 3: Evaluate the Exponential Function

Substitute the exponent back into the exponential function \(e^{x^2-9}\). With the exponent being 0, the expression simplifies to \(e^0\).

Step 4: Determine the Value of \(e^0\)

By definition, any non-zero number raised to the power of 0 is 1. Therefore, \(e^0 = 1\).

Step 5: Conclude the Limit Result

Since as \(x\) approaches 3, the expression \(e^{x^2-9}\) approaches \(e^0\), which is 1, the limit is 1.

Key Concepts

Exponential Functions

Exponential functions are a foundational concept in calculus. They are often expressed in the form \( e^x \), where \( e \) is Euler's number, approximately equal to 2.71828. This number is unique in that it is the base of the natural logarithms, and has special properties in calculus, particularly when dealing with growth and decay processes.
When dealing with exponential functions, you can break down the expression to better understand its components:

• The base \( e \) is constant and forms the backbone of the expression.
• The exponent, in this case, \( x^2 - 9 \), is a function of \( x \). It determines the power to which the base is raised.

Exponential functions grow extremely fast and are used in various real-world applications such as compound interest, population dynamics, and radioactive decay. Understanding them is crucial for evaluating limits especially when the variable approaches a certain point.

Evaluating Limits

In calculus, evaluating limits is the process of finding the value that a function approaches as the operand (often \( x \)) approaches a certain point. This helps us predict the behavior of functions under certain conditions.
For the given problem, we determine the limit of \( \lim_{x \rightarrow 3} e^{x^2-9} \):

• First, identify the behavior of the exponent \( x^2 - 9 \) as \( x \) approaches 3.
• Evaluate the expression \( 3^2 - 9 \) to simplify it.
• Re-calculate the value \( e^0 \) using properties of limits and exponentials.

By simplifying the exponent first, you decrease the complexity of the function. This method—breaking down complicated limits into simpler steps—is a common strategy to make limit evaluation easier, particularly when you face intricate functions.

Exponent Simplification

Exponent simplification is a vital step in solving problems involving exponential functions. By simplifying the exponent before plugging it into the larger expression, you reduce computational complexity.
In the example \( e^{x^2-9} \), simplification involves:

• Substituting 3 for \( x \), resulting in \( 3^2 - 9 \).
• Simplifying \( 3^2 \) to 9, and then calculating \( 9 - 9 = 0 \).

This process reduces the function to \( e^0 \). The result, \( e^0 = 1 \), is derived from the property that any base number raised to the power of zero equals 1. Simplifying the exponent is an effective way to solve calculus problems that might otherwise seem challenging due to their apparent complexity. By focusing on exponent simplification, you ensure the correct and efficient determination of limits.