What is the concentration approaching as t→∞. Please explain

To determine what the concentration \( C(t) \) is approaching as \( t \to \infty \) for the function given in Problem 18, we analyze the function:

\[ C(t) = \frac{200 + 25t}{50 + t} \]

### Step-by-Step Explanation

1. **Understand the Function:**
   The function \( C(t) \) represents the concentration of salt at time \( t \). As \( t \) becomes very large (approaches infinity), we want to understand the behavior of the function.

2. **Simplify the Function for Large \( t \):**
   When \( t \) is very large, the terms with \( t \) will dominate the constants. Therefore, we focus on the terms with \( t \):

   \[ C(t) = \frac{200 + 25t}{50 + t} \]

   We can rewrite this as:

   \[ C(t) = \frac{t(25 + \frac{200}{t})}{t(1 + \frac{50}{t})} \]

3. **Divide Numerator and Denominator by \( t \):**
   Simplify the function by dividing both the numerator and the denominator by \( t \):

   \[ C(t) = \frac{25 + \frac{200}{t}}{1 + \frac{50}{t}} \]

4. **Evaluate the Limits of the Terms:**
   As \( t \) approaches infinity, the terms \( \frac{200}{t} \) and \( \frac{50}{t} \) approach 0:

   \[ \lim_{t \to \infty} \frac{200}{t} = 0 \]
   \[ \lim_{t \to \infty} \frac{50}{t} = 0 \]

5. **Substitute the Limits into the Simplified Function:**
   With the above limits, the function simplifies to:

   \[ C(t) = \frac{25 + 0}{1 + 0} = \frac{25}{1} = 25 \]

### Conclusion
As \( t \to \infty \), the concentration \( C(t) \) approaches 25%. 

### Intuitive Explanation
For very large \( t \), the effect of the initial amounts (200 and 50) becomes negligible compared to the growing influence of the added amounts (25t and t). Essentially, the concentration stabilizes at a value determined by the coefficients of \( t \) in the numerator and the denominator, which are 25 and 1 respectively, giving us a stable concentration of 25% as \( t \) approaches infinity.

Would you like more details or have any other questions?

Here are five questions you might want to ask:
1. How do you find the limit of more complex functions as \( t \to \infty \)?
2. What is the importance of understanding asymptotic behavior in real-world problems?
3. Can you show another example of finding limits for different types of functions?
4. How can we use computational tools to verify these limits?
5. What are common pitfalls to avoid when evaluating limits of functions? 

**Tip:** Always consider the dominant terms in the numerator and denominator when evaluating limits at infinity. This helps to simplify and understand the behavior of the function more clearly.

Explore the concept of limits in mathematics with a focus on concentration approaching infinity. Learn how to determine the concentration as time t approaches infinity for a given function. Suitable for college-level students and learners interested in mathematical analysis.

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