Find the points of discontinuity, and determine whether the...

Find the points of discontinuity, and determine whether the discontinuities are removable. (a) $f(x)=\frac{x^{2}-4}{x^{3}-8}$ (b) $f(x)=\left\{\begin{array}{ll}2 x-3, & x \leq 2 \\ x^{2}, & x>2\end{array}\right.$ (c) $f(x)=\left\{\begin{array}{ll}3 x^{2}+5, & x \neq 1 \\ 6, & x=1\end{array}\right.$

Find the points of discontinuity, and determine whether the discontinuities are removable.
(a) $f(x)=\frac{x^{2}-4}{x^{3}-8}$
(b) $f(x)=\left\{\begin{array}{ll}2 x-3, & x \leq 2 \\ x^{2}, & x>2\end{array}\right.$
(c) $f(x)=\left\{\begin{array}{ll}3 x^{2}+5, & x \neq 1 \\ 6, & x=1\end{array}\right.$

Key Concepts

Removable Discontinuity

A removable discontinuity is a specific type of discontinuity where the limit of the function exists at the point of discontinuity, even though the function is not defined or is defined differently at that point. In such cases, the discontinuity can often be 'removed' by redefining the function's value at that point to match the limit.

Rational Functions

Rational functions are expressions defined by the quotient of two polynomials. When analyzing these functions for discontinuities, common techniques include factoring the polynomials to simplify the expression. Discontinuities may occur where the denominator is zero, and if common factors with the numerator are present, they indicate potential removable discontinuities.

Continuity

Continuity in a function means that at any given point, the limit of the function as it approaches that point equals the function's value at that point. This concept is important because a function that is continuous does not have any breaks, jumps, or holes in its graph.

Discontinuity

A discontinuity occurs when a function fails to be continuous at a point. This may happen due to factors such as undefined points, jumps between differing limit values from each side, or holes in the graph where the function’s value is either not defined or does not match the approaching limit.

Piecewise-defined Functions

Piecewise-defined functions have different expressions based on intervals of the domain. Analyzing their continuity involves checking the behavior of the function at the boundary points between different cases. This typically requires comparing the left-hand limit and the right-hand limit at these boundary points and ensuring they equal the function’s defined value.

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