AIEEE 2002 (Offline)Question 1
$\lim_{x\to\infty}\left(\frac{x^{2}+5x+3}{x^{2}+x+2}\right)^{x}$
Limits & Continuity
Solution
Concept. Base $\to1$, exponent $x\to\infty$. Formulas used. (1) $\lim f^g=e^{\lim g(f-1)}$Step 1 - $f-1$. $\frac{x^2+5x+3}{x^2+x+2}-1=\frac{4x+1}{x^2+x+2}$. Step 2 - Exponent (1). $$\frac{x(4x+1)}{x^2+x+2}=\frac{4+\frac1x}{1+\frac1x+\frac{2}{x^2}}\to4$$ Step 3 - Answer. $e^4$. Answer: A