The given expression is a rational function. To find its limit as x approaches infinity, we divide numerator and denominator by the highest power of x, which is x².

Original: $P=\underset{x\to \infty }{\lim }\frac{x^2+2x+3}{2x^2+3x+4}$

Divide by x²: $\frac{1+\frac{2}{x}+\frac{3}{x^2}}{2+\frac{3}{x}+\frac{4}{x^2}}$

As x→∞, terms with x in the denominator approach 0. The limit simplifies to 1/2.

Final Answer: $\frac{1}{2}$