Question

Let \( f : \mathbb{R} \to \mathbb{R} \) be such that \( f, f', f'' \) are continuous with \( f > 0, f' > 0, f'' > 0. \) Then \( \displaystyle \lim_{x \to -\infty} \frac{f(x) + f'(x)}{2} \) is ............
 

Hint:

For positive, increasing functions with positive derivatives, limits at \( -\infty \) often tend to 0.

Answer 1

Correct Answer: 0

Step 1: Behavior of \( f, f', f'' \).
Since \( f, f', f'' > 0 \), \( f \) is positive and increasing. But as \( x \to -\infty \), typically \( f(x) \to 0 \) for such monotonic positive functions (e.g., exponential).

Step 2: Apply limiting behavior.
As \( x \to -\infty \), both \( f(x) \) and \( f'(x) \) approach 0. Hence, \[ \lim_{x \to -\infty} \frac{f(x) + f'(x)}{2} = \frac{0 + 0}{2} = 0. \]

Final Answer: \[ \boxed{0} \]