https://file.aiquickdraw.com/mathbot/file/1727563223599gub47oqj.jpg

https://file.aiquickdraw.com/mathbot/file/1727563223599fmcq4icz.jpg

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.

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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 \(t \to -\infty\).

Evaluate the limits of the function ℓ(t) = 1 / (1 + e^(-t)) as t approaches 0, ∞, and -∞.

Learn how to evaluate the limits of the exponential function ℓ(t) = 1 / (1 + e^(-t)) as t approaches 0, infinity, and negative infinity. This problem covers key calculus concepts involving limits and the behavior of exponential functions.

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