Question

Use a graphing utility to graph the function and estimate the limit (if it exists). What is the domain of the function? Can you detect a possible error in determining the domain of a function solely by analyzing the graph generated by a graphing utility? Write a short paragraph about the importance of examining a function analytically as well as graphically. $$ \begin{array}{l} f(x)=\frac{x-9}{\sqrt{x}-3} \\ \lim _{x \rightarrow 9} f(x) \end{array} $$

Short answer

The limit of the function as \(x \rightarrow 9\) can be estimated graphically. The domain of the function is \(x \in R \setminus \{9\}\). A possible error in determining the domain of a function solely by analyzing the graph might occur due to the precision limitations of the graphing utility, especially around discontinuities. Both analytical and graphical examinations of a function are similarly important for a thorough understanding of a function.

Step-by-step solution

Graphing the Function

Use a graphing utility to plot the function \( f(x)=\frac{x-9}{\sqrt{x}-3} \). Observation of the graph will provide an estimated value of the limit as \( x \rightarrow 9 \).

Determine the Domain

To find the domain of the function, set the denominator \(\sqrt{x}-3\) not equal to zero. Solving this gives \(x \neq 9\). So, the domain of the function is \(x \in R \setminus \{9\}\).

Analyze Possible Error From Graph

A graphing utility may not always provide a precise value for the domain, especially when there are discontinuities in the function as in this case. Here, the function has a discontinuity at \(x = 9\), but a graphing utility might still depict \(x = 9\) as part of the graph due to the limit of precision.

Importance of Analytical Examination

Analyzing the function analytically as well as graphically is crucial for a precise understanding of the function. While a graphical representation provides visual understanding, it might not accurately depict the nuances like the discontinuities. On the other hand, an analytical representation provides us with a more detailed and accurate understanding, helping us ascertain the precise value of limits, continuity, differentiability, etc. of the function.