Question:
\( \lim_{x \to \infty} \left(\frac{x+5}{x+2}\right)^{x+3} \) equals
\( \lim_{x \to \infty} \left(\frac{x+5}{x+2}\right)^{x+3} \) equals
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$\lim_{x \to \infty} (\frac{x+a}{x+b})^{x+c} = e^{a-b}$. Here $5-2=3$.
Updated On: Apr 10, 2026
- $e$
- $e^2$
- $e^3$
- $e^5$
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The Correct Option is C
Solution and Explanation
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)
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)
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