Q.$\displaystyle\lim_{x\to 0}\left[\frac{\sqrt{x^2+9}-\sqrt{2x^2+9}}{\sqrt{3x^2+4}-\sqrt{2x^2+4}}\right]$

When an expression inside a limit contains a square root (or another radical) and direct substitution produces zero over zero, multiplying both the numerator and the denominator by the radical expression's conjugate clears the troublesome square root using the difference-of-squares identity (a minus b)(a plus b) equals a-squared minus b-squared. This converts the original expression into an equivalent one, valid everywhere x is not equal to a, in which the shared (x minus a) factor can be cancelled cleanly the same way factorization does for polynomials. Some limit …