1. The problem is to evaluate the limit: $$\lim_{n \to \infty} \frac{5}{1+n^2}$$. 2. The formula for limits involving rational functions as $n$ approaches infinity is to analyze the degrees of the numerator and denominator. 3. Here, the numerator is a constant 5, and the denominator is $1+n^2$ which grows without bound as $n$ increases. 4. Since $n^2$ grows much faster than the constant numerator, the fraction approaches zero. 5. Therefore, $$\lim_{n \to \infty} \frac{5}{1+n^2} = 0$$. This means as $n$ becomes very large, the value of the expression gets closer and closer to zero.