Question

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

• A. e
• B. -1
• C. 0
• D. 1

Hint:

Remember: \(e^{kx}\) grows faster than \(x^n\) for any finite \(n\) and \(k > 0\).
So \(\lim_{x \to \infty} \frac{x^n}{e^{kx}} = 0\).

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
The exponential function grows faster than any polynomial function.
As \(x \to \infty\), \(e^{Ax}\) outpaces \(x^r\) for any finite \(r\).

Step 2: Key Formula or Approach:
Use L'Hôpital's rule or the known growth rates.

Step 3: Detailed Explanation:
Consider \(\lim_{x \to +\infty} \frac{x^r}{e^{Ax}}\).
For \(A > 0\), the denominator grows exponentially, while the numerator grows polynomially.
Thus, the limit is 0.
If \(A < 0\), \(e^{Ax} \to 0\), and the limit would be \(\infty\), but the question likely assumes \(A > 0\).
If \(A = 0\), the limit would be \(\infty\) for \(r > 0\).
But the question says \(A\) is any real number and \(r\) is positive.
Typically, such questions assume \(A > 0\) for convergence to 0.
Thus, the limit is 0, which is option (C).
This is a standard result: exponential dominates polynomial.