Give an example of a rational function f(x) that is continuous for all values of x except at x =...

Question:

Give an example of a rational function {eq}f(x) {/eq} that is continuous for all values of {eq}x {/eq} except at {eq}x =\pm 5 {/eq}, where it has non-removable discontinuities, and at {eq}x = 2 {/eq}, where it has a removable discontinuity. Justify your answer using limits.

Removable Discontinuities of a Function:

Suppose {eq}f(x) {/eq} is defined on the intervals {eq}(c-r, c) {/eq} and {eq}(c,c+r) {/eq}. We say that {eq}f(x) {/eq} has a removable discontinuity at {eq}x=c {/eq} if the limit {eq}\lim_{x \to c} f(x) {/eq} exists. This is because the function

{eq}\displaystyle g(x)=\begin{cases} f(x),&x \ne c \\ \lim_{x \to c} f(x), &x = c\end{cases} {/eq}

is continuous at {eq}x=c {/eq} and almost the same as the function {eq}f {/eq}: that is, we can make {eq}f {/eq} be continuous at {eq}x=c {/eq} by defining or redefining {eq}f {/eq} only at the single point {eq}x=c {/eq}.

If {eq}f(x) {/eq} is discontinuous at {eq}x=c {/eq} but does not have a removable discontinuity, we say that {eq}f(x) {/eq} has a non-removable discontinuity.

Answer and Explanation:

A rational function {eq}f(x)=\dfrac{P(x)}{Q(x)} {/eq} will be discontinuous at the zeroes of its {eq}Q(x) {/eq}. So if {eq}R(x) {/eq} has...

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Graphing & Analyzing Rational Functions

from Precalculus: High School

Chapter 13 / Lesson 2
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