lim _(xarrow infty )((2x-3)/(2x+1))^5x

To evaluate the limit \(\lim_{x \to \infty} \left(\frac{2x-3}{2x+1}\right)^{5x}\), we can use the following steps:1. **Simplify the expression inside the limit:** As , the terms and approach 0. Therefore, the expression simplifies to: 2. **Rewrite the limit in terms of the simplified expression:** 3. **Use the approximation as :** 4. **Simplify the exponent:** 5. **Evaluate the limit of each part separately:** - The term grows exponentially as . - The term \((2x+1)^{-5x}\) can be analyzed using Stirling's approximation or the properties of exponential functions. Specifically, \((2x+1)^{-5x}\) behaves like \((2x)^{-5x}\) for large : As , \(\ln(2x) \approx x \ln(2)\), so: Therefore, 6. **Combine the results:** Since grows exponentially and \((2x+1)^{-5x}\) decays exponentially, the overall expression tends to: Thus, the limit is: