Both the top and bottom limits of the first fraction are zero, so we can use l'Hôpital's Rule and take derivatives. Note that 2x is the derivative of x^2-4, and 2x-3 is the derivative of x^2-3x+2. We can then continue to find that
by using the Division Limit Law.
Consider this more complicated example.
As x goes to zero, the limits of cos x - 1 and 3x^2 are both (still) zero, so we can apply l'Hôpital's Rule again.
Note that we used l'Hôpital's Rule twice more in that last line. As long as the limits of the numerator and denominator are still both zero (or both infinte), l'Hôpital's Rule can be used.
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