Understanding Vertical and Removable Discontinuities in Rational Functions.

Rational functions are continous at each point in their domain except when

In P(x)/Q(x) when Q(x)=0

Rational functions are functions that can be written in the form of f(x) = p(x)/q(x), where p(x) and q(x) are polynomials and q(x) ≠ 0. Rational functions are continuous at each point in their domain except when the denominator q(x) is equal to zero.

If q(x) is equal to zero, the function is undefined at that point. This point is called a vertical asymptote, and the function may approach positive infinity, negative infinity, or not approach any finite value at that point. Therefore, rational functions are discontinuous at vertical asymptotes.

For example, consider the rational function f(x) = 1/x. The domain of the function is all real numbers except x = 0. At x = 0, the denominator q(x) is equal to zero, and the function is undefined. Therefore, there is a vertical asymptote at x = 0, and the function is discontinuous at that point.

It is important to note that not all discontinuities of rational functions are vertical asymptotes. Rational functions can also have removable discontinuities, which occur when there is a point where the function is undefined, but the limit of the function at that point exists. In this case, the function’s value can be redefined at that point to make it continuous.

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