Let $f(x) = \frac{1}{1 - e^{x/(1-x)}}$. Which of the following statements regarding the continuity a

Question

Let $f(x) = \frac{1}{1 - e^{x/(1-x)}}$. Which of the following statements regarding the continuity and limits of $f(x)$ is/are CORRECT?

Professor PC Thomas Classes · Answer

The points of discontinuity occur where the denominator is zero or the expression in the exponent is undefined. Setting the denominator to zero, we get $1 - e^{x/(1-x)} = 0$, which implies $\frac{x}{1-x} = 0$, so $x = 0$. Analyzing the limits at $x=0$: as $x 	o 0^+$, $f(x) 	o -\infty$ and as $x 	o 0^-$, $f(x) 	o \infty$. Thus, $x = 0$ is a point of infinite discontinuity (second kind), making (A) correct. The exponent is undefined at $x = 1$. Let us evaluate the limits as $x$ approaches $1$. As $x 	o 1^-$, the term $\frac{x}{1-x} 	o \infty$, so $e^{x/(1-x)} 	o \infty$. Therefore, $\lim_{x 	o 1^-} f(x) = 0$. This makes (C) incorrect. As $x 	o 1^+$, the term $\frac{x}{1-x} 	o -\infty$, so $e^{x/(1-x)} 	o 0$. Therefore, $\lim_{x 	o 1^+} f(x) = \frac{1}{1-0} = 1$, making (D) correct. Since both the left-hand and right-hand limits at $x=1$ exist and are finite but unequal, $x=1$ is a jump discontinuity (first kind), making (B) correct.