Discontinuity Ln 29D16B

1. **State the problem:** We need to determine why the function $f(x) = \ln(x - 27)$ is discontinuous at $a = 27$. 2. **Recall the domain of the natural logarithm:** The function $\ln(x)$ is defined only for $x > 0$. Therefore, $f(x) = \ln(x - 27)$ is defined only when $x - 27 > 0$, or $x > 27$. 3. **Check if $f(27)$ is defined:** Since $f(27) = \ln(27 - 27) = \ln(0)$, and $\ln(0)$ is undefined, the function is not defined at $x = 27$. 4. **Check the limit as $x$ approaches 27:** $$\lim_{x \to 27^+} \ln(x - 27) = \lim_{t \to 0^+} \ln(t) = -\infty$$ The limit from the right side tends to negative infinity. 5. **Check the limit from the left side:** For $x < 27$, $x - 27 < 0$, so $f(x)$ is not defined for $x < 27$. Therefore, the limit from the left does not exist. 6. **Conclusion:** The function is not defined at $x = 27$, and the limit does not exist from the left side. The main reason for discontinuity at $a=27$ is that the function is not defined at that point. **Final answer:** C. The function is not defined at the point given.