To solve problem 8.1, we need to classify the discontinuity points of the function:

$f(x) = e^{\frac{x+1}{x-1}}$

Step-by-Step Solution:

1. Identify Potential Points of Discontinuity: The function $f(x) = e^{\frac{x+1}{x-1}}$ is defined as long as the expression inside the exponent, $\frac{x+1}{x-1}$, is defined. Discontinuities may occur when the denominator of this fraction equals zero, which happens when $x - 1 = 0$, or $x = 1$.

2. Analyze the Point $x = 1$: At $x = 1$, the function $f(x) = e^{\frac{x+1}{x-1}}$ has an undefined expression since $\frac{x+1}{x-1}$ results in a division by zero. Thus, $x = 1$ is a potential point of discontinuity.

3. Classify the Type of Discontinuity: To determine the type of discontinuity, we need to examine the left-hand limit (as $x \to 1^-$) and the right-hand limit (as $x \to 1^+$) of $f(x)$.

   • Left-Hand Limit: As $x \to 1^-$, $\frac{x+1}{x-1} \to -\infty$. Therefore, $e^{\frac{x+1}{x-1}} \to 0$ as $x \to 1^-$.
   • Right-Hand Limit: As $x \to 1^+$, $\frac{x+1}{x-1} \to +\infty$. Consequently, $e^{\frac{x+1}{x-1}} \to +\infty$ as $x \to 1^+$.

   Since the left-hand and right-hand limits do not approach the same value as $x$ approaches 1, we have an infinite discontinuity at $x = 1$.

Conclusion:

The function $f(x) = e^{\frac{x+1}{x-1}}$ has an infinite discontinuity at $x = 1$.

Would you like more details on this classification, or do you have any questions?

Here are five related questions that can help deepen your understanding:

1. How can you determine if a discontinuity is removable?
2. What types of discontinuities exist, and how are they classified?
3. Can a function have more than one type of discontinuity at different points?
4. How does the behavior of exponential functions impact discontinuities?
5. What are some real-world applications where analyzing discontinuities is useful?

Tip: When analyzing discontinuities, always start by identifying points where the function is undefined (like division by zero or logarithm of non-positive values). Then, check the behavior of limits approaching these points to classify the discontinuity type.