what is the limit of the fonction f(x) = $$ \frac{\exponentialE^{x}}{x-1}\text{ when }x\text{ is going to -}\infty $$

To find the limit of the function 

\[
f(x) = \frac{e^x}{x-1}
\]

as $ x \to -\infty $:

1. **Behavior of the numerator $ e^x $:**  
   As $ x \to -\infty $, $ e^x \to 0^+ $ (exponentially fast decay).

2. **Behavior of the denominator $ x - 1 $:**  
   As $ x \to -\infty $, $ x - 1 \to -\infty $.

3. **Combining the behavior:**  
   Since the numerator approaches $ 0 $ and the denominator approaches $ -\infty $, the fraction approaches 

   \[
   \frac{0}{-\infty} = 0.
   \]

Thus, the limit is 

\[
\lim_{x \to -\infty} \frac{e^x}{x - 1} = 0.
\]

Let me know if you'd like further clarifications! 

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### Related Questions:
1. What is the limit of $ \frac{e^x}{x} $ as $ x \to -\infty $?
2. How does $ \frac{e^x}{x-1} $ behave as $ x \to +\infty $?
3. Can $ e^x $ ever approach $ 1 $ or another constant as $ x \to -\infty $?
4. What happens to the function $ \frac{e^x}{x - a} $ for different constants $ a $?
5. How do you find horizontal asymptotes for rational functions involving $ e^x $?

### Tip:
For functions involving $ e^x $, always analyze the growth or decay of $ e^x $ relative to other terms in the function.

What is the limit of the function f(x) = e^x / (x - 1) when x is going to -∞?

Explore the solution to finding the limit of the function f(x) = e^x / (x - 1) as x approaches negative infinity. Understand exponential decay and rational function behavior with a detailed step-by-step explanation. Ideal for Grades 11-12 students.

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