6.114 lim _(xarrow infty )((4x^2+2)/(4x^2)-1)^5x^(2)

To evaluate the limit \(\lim_{x \to \infty} \left( \frac{4x^2 + 2}{4x^2 - 1} \right)^{5x^2}\), we can use the following steps:1. **Rewrite the expression inside the limit:** 2. **Simplify the fraction as approaches infinity:** Therefore, 3. **Substitute this approximation back into the original limit:** 4. **Use the exponential approximation \( (1 + \frac{1}{n})^n \approx e \) as :** Let . Then, As , , so: 5. **Since , we have:** 6. **Evaluate the limit:** Therefore, the limit is: