The problem is asking for the value of $f'(5)$, where the function $f(x)$ satisfies the given limit conditions:

1. $\lim_{x \to \infty} \frac{f(x) - 2x^2}{x^2 - 1} = 2$
2. $\lim_{x \to 1} \frac{f(x) - 2x^2}{x^2 - 1} = 2$

Step 1: Analyze the first limit (as $x \to \infty$)

We are given: $\lim_{x \to \infty} \frac{f(x) - 2x^2}{x^2 - 1} = 2$

This suggests that $f(x)$ behaves like $2x^2$ as $x \to \infty$, but we need to account for any potential lower-degree terms. Let's assume $f(x)$ takes the form: $f(x) = 2x^2 + ax + b$ We now substitute this into the limit expression: $\frac{(2x^2 + ax + b) - 2x^2}{x^2 - 1} = \frac{ax + b}{x^2 - 1}$ As $x \to \infty$, this simplifies to: $\frac{ax + b}{x^2} \approx \frac{a}{x} + \frac{b}{x^2}$ Since the limit is 2, we must have $a = 0$ (as the leading term would otherwise blow up), meaning: $f(x) = 2x^2 + b$ Thus, the first limit tells us that $f(x) = 2x^2 + b$.

Step 2: Analyze the second limit (as $x \to 1$)

We are given: $\lim_{x \to 1} \frac{f(x) - 2x^2}{x^2 - 1} = 2$ Substitute $f(x) = 2x^2 + b$ into the expression: $\frac{(2x^2 + b) - 2x^2}{x^2 - 1} = \frac{b}{x^2 - 1}$ As $x \to 1$, the denominator becomes 0, so for the limit to exist, we use the fact that $x^2 - 1 = (x-1)(x+1)$. Thus: $\lim_{x \to 1} \frac{b}{(x-1)(x+1)} = 2$ Evaluating this limit as $x \to 1$ gives: $\frac{b}{2} = 2 \quad \Rightarrow \quad b = 4$

Step 3: Derive $f'(x)$ and find $f'(5)$

From the above, we have $f(x) = 2x^2 + 4$. Taking the derivative: $f'(x) = 4x$ Thus: $f'(5) = 4 \times 5 = 20$

The value of $f'(5)$ is $\boxed{20}$.

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

Here are five related questions for further understanding:

1. How do limits help determine the form of a function?
2. What happens if the limits were different at infinity and near a specific point?
3. Why is it necessary to account for lower-order terms in function approximations?
4. How does the derivative change with different functional forms?
5. Can you explain how to apply L'Hopital's Rule to limits involving indeterminate forms?

Tip: When working with limits, always look for patterns or terms that dominate the behavior of the function as $x \to \infty$ or near specific points.