Question

Consider the following limit: $ \lim_{\epsilon \to 0} \frac{1}{\epsilon} \int_{0}^{\infty} e^{-x / \epsilon} \left( \cos(3x) + x^2 + \sqrt{x + 4} \right) dx. $ 
Which one of the following is correct?

• A. The limit does not exist.
• B. The limit exists and is equal to \( 0 \).
• C. The limit exists and is equal to \( 3 \).
• D. The limit exists and is equal to \( \pi \).

Hint:

For definite integrals with limits, approximate each term and analyze the leading contributions as the variable approaches the boundary.

Answer 1

The Correct Option is C

1. Simplifying the integral: Using the substitution \( t = x / \epsilon \), we have \( x = \epsilon t \), so \( dx = \epsilon \, dt \). Substituting into the integral, the limit becomes: \[ \lim_{\epsilon \to 0} \frac{1}{\epsilon} \int_{0}^{\infty} e^{-x / \epsilon} \left( \cos(3x) + x^2 + \sqrt{x + 4} \right) dx = \lim_{\epsilon \to 0} \int_{0}^{\infty} e^{-t} \left( \cos(3\epsilon t) + \epsilon^2 t^2 + \sqrt{\epsilon t + 4} \right) dt. \] 2. Breaking into terms: - For the \( \cos(3\epsilon t) \) term: As \( \epsilon \to 0 \), \( \cos(3\epsilon t) \to 1 \). - For the \( \epsilon^2 t^2 \) term: Since \( \epsilon^2 \to 0 \), this term vanishes. - For the \( \sqrt{\epsilon t + 4} \) term: As \( \epsilon \to 0 \), \( \sqrt{\epsilon t + 4} \to 2 \). Substituting these limits into the integral: \[ \int_{0}^{\infty} e^{-t} \left( 1 + 0 + 2 \right) dt = \int_{0}^{\infty} e^{-t} (3) dt. \] 3. Evaluating the integral: \[ \int_{0}^{\infty} 3 e^{-t} dt = 3 \int_{0}^{\infty} e^{-t} dt = 3 \left[ -e^{-t} \right]_{0}^{\infty} = 3 \left( 0 - (-1) \right) = 3. \] Final Answer: \( 3 \).