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12. \( f(x)=\left\{\begin{array}{ll}\frac{x^{2}-9}{x+3}, & x \neq-3 \\ 3, & x=-3\end{array}\right. \) at \( \mathrm{c}=-3 \) In Problems \( 13-20 \), find all points of discontinuity. \( \begin{array}{ll}\text { 13. } g(x)=\frac{\left(x^{2}-1\right)^{2}}{5} & \text { 14. } f(x)=0 \\ \text { 15. } f(x)=\frac{x^{2}+6 x+9}{x^{2}+2 x-15} & \text { 16. } g(x)=\frac{x-3}{x^{2}+x} \\ \begin{array}{ll}\text { 17. } h(x)=\frac{x-7}{x^{3}-x} & \text { 18. } f(x)=\frac{x}{x} \\ \text { 19. } p(x)=\frac{x}{x^{2}+1} & \text { 20. } f(x)=\frac{x^{4}}{x^{4}-1}\end{array}\end{array} \).

Ask by Tyler Evans. in China
Mar 17,2025

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Points of discontinuity: - Problem 12: \( x = -3 \) - Problem 13: None - Problem 14: None - Problem 15: \( x = -5, 3 \) - Problem 16: \( x = 0, -1 \) - Problem 17: \( x = 0, 1, -1 \) - Problem 18: \( x = 0 \) - Problem 19: None - Problem 20: \( x = 1, -1 \)

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In analyzing functions for points of discontinuity, it's always wise to look for values that could make the denominator zero or result in undefined expressions. For instance, in the case of \( f(x) = \frac{x^2 + 6x + 9}{x^2 + 2x - 15} \), factor both the numerator and the denominator; discontinuities may arise when the denominator equals zero, leading to points where the function is not defined. Remember, common mistakes in identifying discontinuities include overlooking removable discontinuities (like holes) or not checking limits from the left and right at key points. When factoring, always double-check your algebra to ensure nothing gets missed; even a small oversight can lead to incorrect conclusions about the behavior of the function at critical points!

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