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. $$ \begin{array}{ll}{\text { (a) } f(x)=\frac{x^{2}-4}{x^{3}-8}} & {\text { (b) } f(x)=\left\{\begin{array}{ll}{2 x-3,} & {x \leq 2} \\ {x^{2},} & {x>2}\end{array}\right.} \\ {\text { (c) } f(x)=\left\{\begin{array}{ll}{3 x^{2}+5,} & {x \neq 1} \\ {6,} & {x=1}\end{array}\right.}\end{array} $$

Find the values of $x$ (if any) at which $f$ is not continuous, and determine whether each such value is a removable discontinuity.
$$
\begin{array}{ll}{\text { (a) } f(x)=\frac{x^{2}-4}{x^{3}-8}} & {\text { (b) } f(x)=\left\{\begin{array}{ll}{2 x-3,} & {x \leq 2} \\ {x^{2},} & {x>2}\end{array}\right.} \\ {\text { (c) } f(x)=\left\{\begin{array}{ll}{3 x^{2}+5,} & {x \neq 1} \\ {6,} & {x=1}\end{array}\right.}\end{array}
$$

Key Concepts

Limit Analysis

Limit analysis is the process of determining the behavior of a function as the input approaches a particular point. This technique is crucial for studying continuity because it allows one to compare the limiting value of a function with its defined value at that point, thereby identifying discontinuities and classifying their types.

Piecewise-Defined Functions

Piecewise-defined functions have different expressions over different parts of their domain. Analyzing discontinuities in these functions involves checking the left-hand limit, right-hand limit, and the function's value at the point where the definition changes. Mismatches among these values can lead to different types of discontinuities, such as jump discontinuities or removable discontinuities.

Continuity

Continuity refers to the property of a function at a point where the limit as the input approaches that point is equal to the function's value at that point. It is a fundamental concept in calculus used to determine whether a function behaves predictably (without sudden jumps or breaks) in a neighborhood around a point.

Removable Discontinuity

A removable discontinuity occurs at a point where the limit of the function exists but the function is either not defined or is defined improperly at that point. Such discontinuities are often identified by the presence of a 'hole' in the graph, which can be 'fixed' by redefining the function at that point so that it equals the limit.

Rational Functions and Factorization

When dealing with rational functions, factorization plays a critical role in detecting points of discontinuity. If the numerator and denominator share a common factor, canceling this factor may reveal a removable discontinuity, since the problematic point might simply be a point where the function was not originally defined rather than an essential singularity.

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