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. $$ \begin{array}{l} f(x)=\frac{x-3}{x^{2}-4 x+3} \\ \lim _{x \rightarrow 3} f(x) \end{array} $$

Short answer

The limit of the function as \(x\) approaches 3 is approximately 1, estimated graphically. The domain is all real numbers except the points where the denominator becomes 0 (upon solving \(x^2 - 4x + 3 = 0\)), as those make the function undefined. Hence, the importance of analyzing a function both graphically and analytically is to avoid possible errors such as missing undefined points(like in this question)that graphical analysis might not show.

Step-by-step solution

Graphing the Function

Use a graphing tool to plot the function \(f(x)=\frac{x-3}{x^{2}-4x+3}\). Observe the graph as \(x\) approaches 3, and estimate the limit.

Estimating Limit from Graph

The limit as \(x\) approaches 3 is the y-value that the graph is approaching as \(x\) gets closer to 3 from both left and right.

Determine Domain from Graph

The domain of a function is the set of all possible values of \(x\) that make the function real and defined. It can be approximately determined from the graph by seeing the \(x\) values for which the function's graph exists.

Analytical Determination of Domain

To find the domain of the function analytically, set the denominator of the function equal to \(0\) and solve: \(x^2 - 4x + 3 = 0\). The solutions will be points where the function is undefined. All other real numbers will form the domain.

Graphical Limitations and Importance of Analytical Analysis

Graphs generated by graphing utilities are based on numerical approximations, so they might not always accurately show the undefined points, leading to potential mistakes in determining the domain. That's why it's essential to examine a function analytically to confirm the findings.