Find Limit \( x \rightarrow \infty \frac{2 \dot{x}^{2}+7 x+5 x}{x^{3}+2 x+1} \)

To find the limit as \( x \) approaches infinity for the expression \( \frac{2 \dot{x}^{2}+7 x+5 x}{x^{3}+2 x+1} \), we first rewrite the numerator: \[ 2 \dot{x}^{2} + 7 x + 5 x = 2 \dot{x}^{2} + 12 x \] Now, we simplify the limit expression: \[ \lim_{x \to \infty} \frac{2 \dot{x}^{2} + 12 x}{x^{3} + 2 x + 1} \] Next, we observe the highest degree terms in the numerator and denominator. Notice that the highest degree term in the numerator depends on \( \dot{x}^{2} \), but there's potentially confusion because \( \dot{x} \) seems to imply a time derivative or another context. Assuming \( \dot{x}^{2} \) is meant to indicate a variable similar to \( x^2 \), we can replace \( \dot{x}^2 \) with \( x^2 \). Then we have: \[ \lim_{x \to \infty} \frac{2x^{2} + 12x}{x^{3} + 2x + 1} \] Now the highest degree term in the numerator is \( 2x^{2} \), and in the denominator, it is \( x^{3} \). Therefore, to compute the limit, we divide all terms by \( x^{3} \): \[ \lim_{x \to \infty} \frac{2x^{2}/x^{3} + 12x/x^{3}}{x^{3}/x^{3} + 2x/x^{3} + 1/x^{3}} \] This simplifies to: \[ \lim_{x \to \infty} \frac{\frac{2}{x} + \frac{12}{x^{2}}}{1 + \frac{2}{x^{2}} + \frac{1}{x^{3}}} \] As \( x \) approaches infinity, both \( \frac{2}{x} \) and \( \frac{12}{x^{2}} \) in the numerator go to 0, and so do \( \frac{2}{x^{2}} \) and \( \frac{1}{x^{3}} \) in the denominator. Thus, we find: \[ \frac{0 + 0}{1 + 0 + 0} = \frac{0}{1} = 0 \] Therefore, the limit is: \[ \boxed{0} \]