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

Use a calculator or other graphing utility to graph the function $g(x)=\frac{x^{2}-2 x+1}{x-1}$. (a) Show that $g(x)$ is not defined at $x=1$. How is this reflected in your calculator graph? (b) Use the graph to argue that even though $g(1)$ is undefined, we have $\lim _{x \rightarrow 1} g(x)=0$

Use a calculator or other graphing utility to graph the function $g(x)=\frac{x^{2}-2 x+1}{x-1}$.
(a) Show that $g(x)$ is not defined at $x=1$. How is this reflected in your calculator graph?
(b) Use the graph to argue that even though $g(1)$ is undefined, we have $\lim _{x \rightarrow 1} g(x)=0$

Key Concepts

Rational Functions

Rational functions are expressions that involve a ratio of two polynomials. They can exhibit interesting behavior such as discontinuities, asymptotes, and holes where the function is not defined, depending on whether factors in the numerator and denominator cancel.

Undefined Points

An undefined point in a function occurs when the function's formula involves a division by zero or some other condition that makes it impossible to compute a value. In the context of rational functions, this often happens when a factor in the denominator equals zero.

Removable Discontinuity

A removable discontinuity is a point at which a function is not defined but at which the limit exists, usually due to the cancellation of a common factor in the numerator and denominator. Although the function might appear to be equivalent to a simpler one everywhere else, the original function does not have a defined value at that point.

Limits

Limits describe the behavior of functions as inputs approach a particular value, even if the function is not defined exactly at that point. They are crucial for understanding the continuity and overall behavior of functions near points of discontinuity.

Graphing Utilities

Graphing utilities are tools such as calculators or software applications that allow users to visualize mathematical functions. When graphed, functions with removable discontinuities typically show a ‘hole’ or gap at the point where the function is undefined, making it easier to understand subtle properties like limits.

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