Problem 36 - A Graphical Approach to Precalculus with Limits

Question

Determine each limit, if it exists. $$\lim _{x \rightarrow 4} 5^{\left(x^{2}-13\right)}$$

Step-by-Step Solution

Answer

The limit is 125.

Step 1: Understand the Limit Expression

We need to evaluate the limit \( \lim _{x \rightarrow 4} 5^{(x^2-13)} \). This involves finding the value that \( 5^{(x^2-13)} \) approaches as \( x \) gets arbitrarily close to 4.

Step 2: Substitute the Limit Value

To find the limit, substitute \( x = 4 \) into the expression \( x^2 - 13 \). This gives us \( (4)^2 - 13 \).

Step 3: Calculate the Inside of the Exponential

Compute \( (4)^2 - 13 = 16 - 13 = 3 \). So, the exponent becomes 3.

Step 4: Evaluate the Exponential Function

Now that we have the exponent as 3, we find \( 5^3 = 125 \).

Step 5: Conclusion About the Limit

Since substitution didn't result in an indeterminate form and all calculations are finite, the limit exists and is \( 125 \).

Key Concepts

Exponential Functions

Exponential functions are a crucial part of calculus and many other areas in mathematics. At their core, exponential functions have a constant base raised to a variable exponent. For example, in the function \(f(x) = 5^{x}\), the base is 5 and the exponent is \(x\). This differs from polynomial functions where the variable is the base and the power is constant.
An essential feature of exponential functions is their rapid growth rate compared to polynomial and linear functions. This makes them significant in modeling real-world phenomena such as population growth, radioactive decay, and compound interest. Understanding how these functions behave is key, especially when analyzing limits, as seen in the original exercise.
When dealing with limits involving exponential functions, the main challenge is often evaluating how changes in the input affect the function's output, such as in the case of \(5^{(x^2-13)}\). Keeping in mind the properties of exponents can help solve these problems effectively.

Evaluating Limits

The concept of limits is fundamental in calculus and forms the groundwork for much of the analysis in this branch of mathematics. Evaluating a limit involves finding the value that a function approaches as the input approaches a certain point.
In this scenario, we're looking at \(\lim _{x \rightarrow 4} 5^{(x^2-13)}\). We are interested in what happens to \(5^{(x^2-13)}\) as \(x\) closes in on 4. Limits help us to understand the behavior of functions near specific points, even if the function is not defined at that point.
One important aspect is the distinction between finite limits, which result in a specific value, and limits that do not exist if the function behaves erratically or grows without bound as it approaches the point.

Substitution Method

The substitution method is a straightforward approach for evaluating limits when the function is continuous at the point of interest. This technique involves directly substituting the value into the function. If this does not result in an indeterminate form, such as \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\), it often gives the limit.
In the problem \(\lim _{x \rightarrow 4} 5^{(x^2-13)}\), by substituting \(x = 4\), we calculate the expression inside the exponent. Substituting directly gives us \((4)^2 - 13 = 3\). Hence, the problem reduces to evaluating \(5^3\), which is 125.
The simplicity of this method makes it very effective for many calculus problems, especially when combined with the properties of exponential functions. However, caution is needed since some problems might still result in indeterminate forms requiring further techniques or transformations to resolve.