The second image shows the function $\ell(t)$ as:

$\ell(t) = \frac{1}{1 + e^{-t}}$

We are tasked with evaluating the following limits for $\ell(t)$:

1. $\lim_{t \to 0} \ell(t)$
2. $\lim_{t \to \infty} \ell(t)$
3. $\lim_{t \to -\infty} \ell(t)$

Solving the limits

1. $\lim_{t \to 0} \ell(t)$: Substitute $t = 0$ into $\ell(t)$: $\ell(0) = \frac{1}{1 + e^{0}} = \frac{1}{1 + 1} = \frac{1}{2}$ Answer: $\frac{1}{2}$

2. $\lim_{t \to \infty} \ell(t)$: As $t \to \infty$, $e^{-t} \to 0$, so: $\ell(t) = \frac{1}{1 + e^{-t}} \to \frac{1}{1 + 0} = 1$ Answer: $1$

3. $\lim_{t \to -\infty} \ell(t)$: As $t \to -\infty$, $e^{-t} \to \infty$, so: $\ell(t) = \frac{1}{1 + e^{-t}} \to \frac{1}{1 + \infty} = 0$ Answer: $0$

Let me know if you want any additional details or further clarification.

Here are some follow-up questions to extend this topic:

1. What is the derivative of $\ell(t)$ with respect to $t$?
2. How does the function $\ell(t)$ behave for large positive and negative values of $t$?
3. Can you graph the function $\ell(t)$ and its asymptotes?
4. What is the second derivative of $\ell(t)$, and what does it tell us about the concavity of the function?
5. How would you interpret $\ell(t)$ in the context of logistic growth?

Tip: When evaluating limits involving exponentials, always consider how $e^t$ behaves as $t \to \infty$ and $t \to -\infty$.