Let's analyze the question step by step.

The function provided is:

$h(x) = \frac{1}{e^x + 10}.$

Question 1: Determining the limit of $e^x$ as $x \to -\infty$

As $x \to -\infty$:

• The exponential $e^x$ approaches 0 because the exponential function decays rapidly for large negative values of $x$.

$\lim_{x \to -\infty} e^x = 0.$

Question 2: Determining the limit of $e^x + 10$ as $x \to -\infty$

From above, since $e^x \to 0$, we get: $e^x + 10 \to 10 \quad \text{as } x \to -\infty.$

Thus: $\lim_{x \to -\infty} (e^x + 10) = 10.$

Question 3: Determining the limit of $h(x)$ as $x \to -\infty$

Substituting $e^x + 10 \to 10$ into $h(x)$, we calculate: $\lim_{x \to -\infty} h(x) = \frac{1}{e^x + 10} = \frac{1}{10}.$

So: $\lim_{x \to -\infty} h(x) = \frac{1}{10}.$

Question 4: Determining the limit of $h(x)$ as $x \to +\infty$

As $x \to +\infty$:

• $e^x \to +\infty$, so $e^x + 10 \to +\infty$.

Substituting into $h(x)$: $\lim_{x \to +\infty} h(x) = \frac{1}{e^x + 10} = 0.$

So: $\lim_{x \to +\infty} h(x) = 0.$

Question 5: Finding the variation table of $h(x)$

1. Derivative of $h(x)$:

The derivative of $h(x)$ is calculated using the quotient rule: $h'(x) = \frac{-e^x}{(e^x + 10)^2}.$

• The numerator $-e^x$ is negative for all $x$, and the denominator $(e^x + 10)^2$ is always positive. Therefore, $h'(x) < 0$ for all $x$.

2. Monotonicity:

Since $h'(x) < 0$, $h(x)$ is strictly decreasing for all $x \in \mathbb{R}$.

3. Limits:

From the above calculations:

• As $x \to -\infty$, $h(x) \to \frac{1}{10}$.
• As $x \to +\infty$, $h(x) \to 0$.

Variation Table

I will now create a clear and pd import ace_tools as tools

Creating the variation table for h(x)

data = { "x": ["-∞", "∞"], "h(x)": ["1/10", "0"], "Monotonicity": ["Decreasing", "Decreasing"] }

variation_table = pd.DataFrame(data) tools.display_dataframe_to_user(name="Variation Table for h(x)", dataframeI have provided a clear variation table for the function $h(x)$. Let me know if you need a graphical representation or further details.

Here are five related questions to expand your understanding:

1. How would the variation table change if $h(x)$ were increasing instead of decreasing?
2. Can you explain why the exponential $e^x$ dominates the behavior of $h(x)$ as $x \to +\infty$?
3. How do asymptotes relate to the limits calculated in this problem?
4. What happens to the derivative $h'(x)$ at extreme values of $x$?
5. How can you extend this analysis to functions of the form $h(x) = \frac{1}{e^x + c}$?

Tip: Always check the