Question

Graphical, Numerical, and Analytic Analysis In Exercises \(75-82\) , use a graphing utility to graph the function and estimate the limit. Use a table to reinforce your conclusion. Then find the limit by analytic methods. $$ \lim _{x \rightarrow 16} \frac{4-\sqrt{x}}{x-16} $$

Short answer

The limit of the function \(f(x) = \frac{4-\sqrt{x}}{x-16}\) as \(x\) approaches 16 is \(-\frac{1}{8}\).

Step-by-step solution

Graphical Analysis

Use a graphing utility to graph the function \(f(x) = \frac{4-\sqrt{x}}{x-16}\). Observe the behavior of the function as \(x\) approaches 16 from both left and right sides. You can see a hole, or a point discontinuity, at \(x=16\) which indicates the function is undefined at \(x=16\).

Numerical Analysis

Create a table of values that encodes the behavior of \(f(x)\) as \(x\) approaches 16 from both left and right sides. From the values in the table, infer the estimated limit.

Analytical Analysis

To find the exact limit, apply L'Hopital's rule, which states that the limit of a quotient of two functions as \(x\) approaches a certain value can be found by taking the derivative of the numerator and the derivative of the denominator. Here the numerator is \(4-\sqrt{x}\) and the denominator is \(x-16\). The derivative of the numerator is \(-\frac{1}{2\sqrt{x}}\) and the derivative of the denominator is 1. Therefore, the limit as \(x -> 16\) is \(\lim_{x \rightarrow 16} \frac{-\frac{1}{2\sqrt{x}}}{1} = -\frac{1}{8}\).

Key concepts

Graphical Analysis of Limits

Understanding the behavior of functions as they approach a specific input can be achieved through graphical analysis. In calculus, graphical analysis involves observing how the function looks on a graph as it gets closer to a certain point. This visual inspection can provide insights into limits, continuity, and potential asymptotes.

For the function \(f(x) = \frac{4-\sqrt{x}}{x-16}\), when graphed, you may see that as \(x\) approaches 16, the function doesn't have a value exactly at \(x=16\) but seems to approach a certain value just before and after 16. This visual cue is a 'hole' on the graph and it suggests the presence of a limit that we're interested in estimating. Using a graphing utility not only aids in estimating this limit but also reinforces the concept that although a function may not be defined at a point, limits can still exist around that point.

Numerical Analysis of Limits

With numerical analysis, you can approximate the limit of a function by constructing a table of values. This method closes in on the problematic point by selecting values of \(x\) that get increasingly closer to the number in question from both directions.

For example, to investigate \(\lim_{x \rightarrow 16} \frac{4-\sqrt{x}}{x-16}\), you would select values around 16, such as 15.9, 15.99, 16.01, and 16.1, and observe the resulting function values. If these results consistently approach a single number from both sides of 16, this suggests that the limit is that number. Numerical analysis provides concrete evidence of what the function is doing near the point of interest, making it a valuable tool for understanding limits.

L'Hopital's Rule

When functions become complex, and direct substitution is not possible due to indeterminate forms like \(0/0\), L'Hopital's Rule comes to the rescue. It's a sophisticated method that assists in finding the limit by differentiating the numerator and the denominator separately.

In the function \(f(x) = \frac{4-\sqrt{x}}{x-16}\), direct substitution of \(x=16\) gives us \(0/0\), an indeterminate form. According to L'Hopital's Rule, we take derivatives of the top and bottom until we can evaluate the limit directly. After differentiating, as seen in the provided solution, the limit as \(x\) approaches 16 is calculated with the derivatives, leading to \(\lim_{x \rightarrow 16} \frac{-\frac{1}{2\sqrt{x}}}{1} = -\frac{1}{8}\). L'Hopital’s Rule thus simplifies the process of finding limits in more challenging scenarios.

Analytic Methods in Calculus

Analytic methods take a deep dive into the workings of functions without relying solely on numerical or graphical interpretations. This approach involves mathematical techniques, theorems, and algebraic manipulations to find limits and understand function behavior.

In the case of our function \(f(x) = \frac{4-\sqrt{x}}{x-16}\), aside from L'Hopital's Rule, we might attempt to simplify the expression algebraically by multiplying by a conjugate or using expansion to eliminate the problematic term causing the indeterminate form. Analytic methods require a thorough understanding of the properties of limits and the underlying algebra to be effective. By using these techniques, students gain a deeper insight into calculus and the behavior of functions, going beyond what might be immediately visible through graphs or tables.