Evaluate the limit (3x+5)/(x^2+1), x->+infinity?

In order to calculate the limit of a rational function, when x tends to +inf., we'll divide both, numerator and denominator, by the highest power of x, which in this case is x^2.

We'll have:

lim (3x+5)/(x^2+1) = lim (3x+5)/lim (x^2+1)

lim (3x+5)/lim (x^2+1) = lim x^2*(3/x + 5/x^2)/lim x^2*(1 + 1/x^2)

After reducing similar terms, we'll get:

lim (3x+5)/lim (x^2+1) = (0+0)/(1+0)=0