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We are tasked with finding \(\lim_{{x \to \infty}} f(x)\), given that for all \(x > 1\), the function \(f(x)\) satisfies the inequality:

\[
\frac{10e^x - 23}{2e^x} < f(x) < \frac{5\sqrt{x}}{\sqrt{x-1}}.
\]

### Step 1: Investigate the left-hand side limit as \(x \to \infty\)
The left-hand side expression is:

\[
\frac{10e^x - 23}{2e^x} = \frac{10e^x}{2e^x} - \frac{23}{2e^x} = 5 - \frac{23}{2e^x}.
\]

As \(x \to \infty\), the term \(\frac{23}{2e^x} \to 0\) because \(e^x\) grows exponentially. Therefore:

\[
\lim_{{x \to \infty}} \left( 5 - \frac{23}{2e^x} \right) = 5.
\]

### Step 2: Investigate the right-hand side limit as \(x \to \infty\)
Now let's examine the right-hand side expression:

\[
\frac{5\sqrt{x}}{\sqrt{x-1}}.
\]

For large \(x\), we can approximate \(\sqrt{x - 1} \approx \sqrt{x}\). Hence:

\[
\frac{5\sqrt{x}}{\sqrt{x-1}} \approx \frac{5\sqrt{x}}{\sqrt{x}} = 5.
\]

Thus, as \(x \to \infty\):

\[
\lim_{{x \to \infty}} \frac{5\sqrt{x}}{\sqrt{x-1}} = 5.
\]

### Step 3: Apply the Squeeze Theorem
We now have the following situation as \(x \to \infty\):

\[
5 \leq f(x) \leq 5.
\]

By the Squeeze Theorem, since both bounds approach 5, we conclude that:

\[
\lim_{{x \to \infty}} f(x) = 5.
\]

Thus, the limit of \(f(x)\) as \(x \to \infty\) is 5.

Would you like more details on any step, or do you have any further questions?

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Here are five related questions to explore:

1. What is the Squeeze Theorem, and how does it apply in this case?
2. How do exponential functions behave as \(x \to \infty\)?
3. How do you approximate square roots of large numbers, like \(\sqrt{x - 1}\)?
4. Can you apply the Squeeze Theorem to other types of inequalities?
5. What would happen if the limits of the upper and lower bounds were not equal?

**Tip:** Always check the behavior of functions like exponentials and roots as \(x \to \infty\), as they often simplify in the limit.

Find lim f(x) as x approaches infinity given the inequality: (10e^x - 23) / (2e^x) < f(x) < (5√x) / (√x - 1) for x > 1.

This problem involves finding the limit of a function using the Squeeze Theorem. Learn how to handle inequalities with exponential and square root functions and apply the theorem to find the limit as x approaches infinity.

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