1. Problem statement: Evaluate the limit $\lim_{x\to+\infty} (x^2+3x+5)$. 2. Idea: Recognize that the polynomial is a parabola that opens upward and the highest power $x^2$ dominates as $x\to+\infty$. 3. Factor out $x^2$ to see the dominant behavior. $$x^2+3x+5 = x^2\left(1+\frac{3}{x}+\frac{5}{x^2}\right)$$ 4. Evaluate limits of the small terms: As $x\to+\infty$ we have $\frac{3}{x}\to 0$ and $\frac{5}{x^2}\to 0$. 5. Therefore the factor in parentheses tends to $1$ and the whole expression behaves like $x^2$ which tends to $+\infty$ as $x\to+\infty$. 6. Conclude the limit: $\lim_{x\to+\infty} (x^2+3x+5) = +\infty$.