Q.$\displaystyle\lim_{x\to \infty}\left[\frac{\sqrt{x^2+5}-\sqrt{x^2-3}}{\sqrt{x^2+3}-\sqrt{x^2+1}}\right]$

This describes what happens to a function's value as the input itself grows without any upper bound (x tends to positive infinity) or without any lower bound (x tends to negative infinity), as opposed to the input approaching some fixed finite point. For the simple function one over x, as x grows arbitrarily large in either the positive or the negative direction, the value one over x shrinks arbitrarily close to zero — the standard example behind the formal definition, which requires that for every tolerance epsilon there be some threshold M beyond which the function's value stays within epsilon of the limiting value l. For a ratio of polynomials (or any expression that grows without bound in both its numerator and its denominator), the standing technique is to divide every term, top and bottom, by the highest power of x present in the whole expression; every leftover term of the form a constant over a positive power of x then vanishes in the limit, leaving only the ratio of the leading coefficients — or, if the degrees of numerator and denominator differ, an ou …