Question

If \(A\) is any real number and \(r\) is a positive rational number, then the value of \(\lim_{x\to +\infty}\frac{A}{x^r}\) is:

• A. e
• B. 1
• C. $-1$
• D. 0

Hint:

Any expression of the form $\frac{\text{Constant}}{\infty}$ always evaluates to $0$ in limit evaluations.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
This problem requires evaluating the limit of a rational function as the independent variable approaches positive infinity.
Key Formula or Approach:
For any constant $A \in \mathbb{R}$ and exponent $r > 0$: \[ \lim_{x\to +\infty} x^r = +\infty \] Therefore: \[ \lim_{x\to +\infty} \frac{1}{x^r} = 0 \]

Step 2: Detailed Explanation:
Let us apply the limit properties to the given expression: \[ L = \lim_{x\to +\infty} \frac{A}{x^r} \] Since $A$ is a constant real number, we can pull it out of the limit: \[ L = A \cdot \lim_{x\to +\infty} \frac{1}{x^r} \] Because $r$ is a positive rational number ($r > 0$), as $x$ grows infinitely large, $x^r$ also grows infinitely large.
Thus, the reciprocal $\frac{1}{x^r}$ approaches zero: \[ L = A \cdot 0 = 0 \] Hence, the limit value is 0.

Step 3: Final Answer:
The limit is 0, which corresponds to Option (D).