18. [-11 Points] DETAILS MY NOTES SCA Find \( \lim _{x \rightarrow \infty} f(x) \) if, for all \( x>1 \), \( \frac{8 e^{x}-25}{2 e^{x}}

Ask by Pollard Lynch. in the United States

Feb 26,2025

To find \( \lim_{x \rightarrow \infty} f(x) \), we need to analyze the behavior of the bounds given for \( f(x) \). 1. First, consider the upper bound: \[ \frac{4 \sqrt{x}}{\sqrt{x-1}}. \] As \( x \) approaches infinity, we can simplify this expression: \[ \sqrt{x-1} \approx \sqrt{x}, \] thus, \[ \frac{4 \sqrt{x}}{\sqrt{x-1}} \approx \frac{4 \sqrt{x}}{\sqrt{x}} = 4. \] Therefore, \[ \lim_{x \rightarrow \infty} \frac{4 \sqrt{x}}{\sqrt{x-1}} = 4. \] 2. Now, consider the lower bound: \[ \frac{8 e^{x}-25}{2 e^{x}}. \] As \( x \) approaches infinity, \( 25 \) becomes negligible compared to \( 8 e^{x} \): \[ \frac{8 e^{x}-25}{2 e^{x}} \approx \frac{8 e^{x}}{2 e^{x}} = 4. \] Thus, \[ \lim_{x \rightarrow \infty} \frac{8 e^{x}-25}{2 e^{x}} = 4. \] Since \( f(x) \) is squeezed between these two limits, by the Squeeze Theorem: \[ \lim_{x \rightarrow \infty} f(x) = 4. \] In conclusion: \[ \lim _{x \rightarrow \infty} f(x) = 4. \]