https://r2.erweima.ai/mathbot/image/1726092623268cw27dbsp.jpg

hallar el valor de a y b para que el limite de ese resultado

The limit problem you've provided is:

\[
\lim_{x \to +\infty} \left( \frac{2x + 3a}{2x + b} \right)^x = e^4
\]

We want to find the values of \( a \) and \( b \) such that the limit holds.

### Step-by-Step Approach

1. **Behavior of the fraction as \( x \to \infty \):**

   As \( x \) grows large, both the numerator and denominator are dominated by \( 2x \). So, we can approximate the fraction as:

   \[
   \frac{2x + 3a}{2x + b} \approx 1 + \frac{3a - b}{2x}
   \]

2. **Limit expression for exponential growth:**

   We use the fact that for a small \( u \), \( (1 + u)^x \) approximates \( e^{ux} \) as \( x \to \infty \). Applying this to the expression:

   \[
   \left( 1 + \frac{3a - b}{2x} \right)^x \to e^{3a - b}
   \]

3. **Equating to the given limit:**

   The limit provided is:

   \[
   e^{3a - b} = e^4
   \]

   Therefore, we can equate the exponents:

   \[
   3a - b = 4
   \]

4. **Conclusion:**

   The values of \( a \) and \( b \) must satisfy the equation:

   \[
   3a - b = 4
   \]

   Therefore, \( b = 3a - 4 \). There is no unique solution unless further information is provided. For example, if one of the variables is specified, the other can be determined.

Would you like to explore this further or get details on specific steps?

### Relative Questions:
1. How do approximations for limits work at infinity?
2. What other methods can be used to solve limits involving exponential expressions?
3. Can you always approximate fractions for limits at infinity like in this case?
4. How does the expansion \( (1 + u)^x \approx e^{ux} \) work for different values of \( u \)?
5. What are other examples where limits result in exponential forms?

**Tip:** When dealing with limits involving powers, especially as \( x \) approaches infinity, logarithmic transformations and approximations can often simplify the problem.

Hallar el valor de a y b para que el límite de ese resultado

This calculus problem involves determining the values of a and b such that the limit of (2x + 3a)/(2x + b)^x as x approaches infinity equals e^4. It covers concepts like limits, exponential functions, and asymptotic behavior. Suitable for college-level calculus students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation