The answer is Zero, but I suspect you knew that before you posed the question.

We can simplify this problem without resorting to calculus. Recall that

lim x->infty(f(x) + g(x)) = lim x->infty(f(x)) + lim x->infty(g(x))

Now it is easy to see that sqrt(x^2+5) approaches infinity with x, and so does sqrt(x^2+3). Since the limits are equal, the difference of the limits is zero.

**Further Reading**

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