Question

\( \lim_{x \to \infty} \left(\frac{x+5}{x+2}\right)^{x+3} \) equals

• A. $e$
• B. $e^2$
• C. $e^3$
• D. $e^5$

Hint:

$\lim_{x \to \infty} (\frac{x+a}{x+b})^{x+c} = e^{a-b}$. Here $5-2=3$.

Answer 1

The Correct Option is C

Step 1: Simplify Expression
The limit is in the $1^{\infty}$ form. $\lim_{x \to \infty} \left(\frac{x+2+3}{x+2}\right)^{x+3} = \lim_{x \to \infty} \left(1 + \frac{3}{x+2}\right)^{x+3}$.
Step 2: Apply Standard Limit
Use $\lim_{x \to \infty} (1 + f(x))^{g(x)} = e^{\lim_{x \to \infty} f(x)g(x)}$. Power $= \lim_{x \to \infty} \frac{3(x+3)}{x+2} = \lim_{x \to \infty} \frac{3(1 + 3/x)}{1 + 2/x} = 3$.
Step 3: Conclusion
The value is $e^3$.
Final Answer: (c)