Given the expression:

$\lim_{{x \to -\infty}} \left( 1 - \frac{x+1}{e^x} \right)$

Let's analyze it with $x$ approaching $-\infty$.

Analyzing the Term $\frac{x+1}{e^x}$:

• As $x \to -\infty$, $e^x \to 0^+$ (since the exponential function approaches 0 as its argument goes to $-\infty$).
• The term $x+1$ will approach $-\infty$ as $x \to -\infty$.

Behavior of the Term:

• $\frac{x+1}{e^x}$ becomes $\frac{-\infty}{0^+}$, which tends to $-\infty$.

Thus, the original expression can be analyzed as:

$\lim_{{x \to -\infty}} \left( 1 - \frac{x+1}{e^x} \right)$

This limit can be seen as:

$\lim_{{x \to -\infty}} \left( 1 - (-\infty) \right)$

Simplifying:

The expression simplifies to:

$1 - (-\infty) = 1 + \infty = \infty$

Therefore, the limit is:

$\boxed{\infty}$

Would you like any details or have any questions?

Here are 8 relative questions that expand on the information:

1. How does the behavior of $e^x$ change as $x$ approaches $-\infty$?
2. Why does $\frac{x+1}{e^x}$ tend to $-\infty$ as $x$ approaches $-\infty$?
3. Can you explain the difference in behavior of exponential functions as $x \to \infty$ versus $x \to -\infty$?
4. What are the implications of an expression tending to $\infty$ in a limit?
5. How can we use the concept of asymptotes to understand the behavior of functions as $x \to \pm \infty$?
6. How does the presence of $x+1$ in the numerator affect the limit compared to $x$ alone?
7. Can you give an example of a limit involving exponential functions that approaches a finite value?
8. How do we handle limits involving $e^x$ and other functions as $x \to -\infty$?

Tip: When analyzing limits involving exponential functions, remember that $e^x$ grows very quickly as $x \to \infty$ and decays very quickly as $x \to -\infty$.