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Chapter 1: Problem 47

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

Expert verified
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

01

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 \( x \rightarrow 9 \).
02

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 \(x \in R \setminus \{9\}\).
03

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.
04

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.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Graphical Analysis
In calculus, graphical analysis is a fundamental tool used to visually explore the behavior of functions. By using a graphing utility, we can generate the graph of a function and examine its shape and behavior as input values change. For instance, when graphing the function \( f(x)=\frac{x-9}{\sqrt{x}-3} \), we can visually investigate how the function acts as \( x \) approaches 9. However, graphical analysis does come with its imperfections.
Graphs can provide estimates but might not reveal underlying algebraic facts, such as asymptotic behavior or specific limits.
  • Graphs help in visually identifying trends and behavior patterns of functions.
  • They can imply points of discontinuity but may not show the exact nature.
Thus, graphical analysis should be used alongside analytical methods to confirm what the graph suggests.
Domain of a Function
The domain of a function refers to the set of all possible input values \( x \) for which the function is defined. For the function \( f(x)=\frac{x-9}{\sqrt{x}-3} \), identifying the domain involves ensuring that the expressions within the function are valid. Here, the denominator \( \sqrt{x}-3 \) should not be zero, which means \( x eq 9 \). Hence, the domain of this function is all real numbers except for 9:
  • \( x \in \mathbb{R} \setminus \{9\} \)
Determining a function's domain analytically ensures accuracy, as graphical representations may misrepresent or overlook singularities or asymptotic restrictions.
Discontinuities in Functions
Discontinuities occur at points where a function fails to be continuous, which can manifest as jumps, holes, or vertical asymptotes in the graph. For the given function, \( f(x)=\frac{x-9}{\sqrt{x}-3} \), there is a removable discontinuity at \( x=9 \) because the function is undefined there due to the zero in the denominator. A graphing utility might display this differently based on precision,
  • Graph might visually link points where discontinuity occurs.
  • Important to analytically recognize these areas to avoid misconceptions.
Analytical examination helps to correctly identify and understand discontinuities of a function, ensuring that calculations involving limits and continuity are accurately interpreted.

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Most popular questions from this chapter

Find all values of \(c\) such that \(f\) is continuous on \((-\infty, \infty)\). \(f(x)=\left\\{\begin{array}{ll}1-x^{2}, & x \leq c \\ x, & x>c\end{array}\right.\)

In the context of finding limits, discuss what is meant by two functions that agree at all but one point.

Write the expression in algebraic form. \(\csc \left(\arctan \frac{x}{\sqrt{2}}\right)\)

Prove that if \(f\) has an inverse function, then \(\left(f^{-1}\right)^{-1}=f\).

Sketch the graph of the function. Use a graphing utility to verify your graph. $$ f(x)=\operatorname{arcsec} 2 x $$
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