Use a calculator or other graphing utility to graph the...

Use a calculator or other graphing utility to graph the function $f(x)=\frac{x-2}{x^{2}-x-2}$. (a) Show that $f(x)$ is not defined at $x=2$. How is this reflected in your calculator graph? (b) Use the graph to argue that even though $f(2)$ is undefined, we have $\lim _{x \rightarrow 2} f(x)=\frac{1}{3}$.

Use a calculator or other graphing utility to graph the function $f(x)=\frac{x-2}{x^{2}-x-2}$.
(a) Show that $f(x)$ is not defined at $x=2$. How is this reflected in your calculator graph?
(b) Use the graph to argue that even though $f(2)$ is undefined, we have $\lim _{x \rightarrow 2} f(x)=\frac{1}{3}$.

Key Concepts

Domain of a Function

The domain of a function consists of all x-values for which the function is defined. In rational functions, values of x that make the denominator zero must be excluded from the domain. Recognizing these undefined points is crucial when analyzing the behavior of the function, as they often lead to discontinuities or asymptotic behavior in the graph.

Graphing Utility

Graphing utilities or calculators allow us to plot the behavior of functions, highlighting features such as discontinuities, asymptotes, and removable holes. These tools can visually identify points where a function is undefined, by showing gaps or breaks in the graph, even if the algebraic expression simplifies to suggest a continuous behavior elsewhere.

Limits of Functions

The concept of limits involves investigating the behavior of a function as the input value approaches a specific point, regardless of whether the function is defined at that point. Limits provide insight into the trend of the function’s outputs near points of interest, which is particularly important when dealing with discontinuities, as the limit may exist even when the function's value is undefined.

Removable Discontinuity

A removable discontinuity occurs when a function is undefined at a specific point, often because of a common factor in the numerator and denominator cancelling out, yet the limit as x approaches that point exists. This situation suggests that the 'hole' in the graph at that point does not affect the overall behavior of the function nearby, and the limit indicates what the value of the function would be if the discontinuity were 'filled in'.

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