Question

\[ \lim_{x \to \infty} 5 \left( 1 + \frac{4}{x} \right)^x = \]

• A. 5
• B. \( 5e \)
• C. \( 5e^4 \)
• D. 0

Hint:

Recognize the standard limit form \( \left( 1 + \frac{a}{x} \right)^x \) as it approaches \( e^a \) as \( x \to \infty \).

Answer 1

The Correct Option is C

We recognize that the expression \( \left( 1 + \frac{4}{x} \right)^x \) is a form of the limit definition of the exponential function \( e \). As \( x \to \infty \), we have: \[ \lim_{x \to \infty} \left( 1 + \frac{4}{x} \right)^x = e^4. \] Thus, the limit becomes: \[ \lim_{x \to \infty} 5 \left( 1 + \frac{4}{x} \right)^x = 5e^4. \]