Find the values of x (if any) at which f is not continuous,...

Find the values of $x$ (if any) at which $f$ is not continuous, and determine whether each such value is a removable discontinuity. $$ \begin{array}{ll}{\text { (a) } f(x)=\frac{|x|}{x}} & {\text { (b) } f(x)=\frac{x^{2}+3 x}{x+3}} \\ {\text { (c) } f(x)=\frac{x-2}{|x|-2}}\end{array} $$

Find the values of $x$ (if any) at which $f$ is not continuous, and determine whether each such value is a removable discontinuity.
$$
\begin{array}{ll}{\text { (a) } f(x)=\frac{|x|}{x}} & {\text { (b) } f(x)=\frac{x^{2}+3 x}{x+3}} \\ {\text { (c) } f(x)=\frac{x-2}{|x|-2}}\end{array}
$$

Key Concepts

Removable Discontinuity

A removable discontinuity is a type of discontinuity that occurs when a function is not defined or does not match the limit at a particular point, despite the limit existing. This is often due to a common factor in the numerator and denominator of a rational function that can be canceled out. By redefining the function at the point of discontinuity, the function can be 'fixed' to become continuous.

Discontinuity

A discontinuity of a function occurs at a point where the function is not continuous, meaning the limit exists but does not equal the function’s value, or the limit does not exist at all. Discontinuities can be classified into various types, such as removable and non?removable discontinuities, each with different characteristics and implications for the function’s behavior.

Algebraic Simplification

Algebraic simplification involves manipulating expressions to cancel factors, factor polynomials, or simplify complex fractions. This technique is especially useful in the analysis of rational functions, as it can reveal hidden behaviors such as removable discontinuities by canceling out common factors that cause undefined expressions at specific points.

Continuity

Continuity is a fundamental concept in calculus that refers to a function having no breaks, jumps, or holes at a point or over an interval. A function is continuous at a point if the function's limit as it approaches the point is equal to the function's value at that point. Understanding continuity is crucial for analyzing the behavior of functions and applying many important theorems in calculus.

Limits

Limits are used to describe the behavior of a function as its input approaches a particular value. They are essential in understanding functions at points where they may not be well-defined or where direct substitution leads to an indeterminate form. Evaluating one?sided limits (approaching from the left or the right) can help in detecting discontinuities and understanding the nature of these discontinuities.