L'Hospitals Rule - Introduction

Suppose f and g are two real functions which are defined in an open interval containing a real Number a, and we whish to compute the limit

Sometimes the Answer is easy. Assume that the limits of f(x) and g(x) exist as x→a,

lim_x→a f(x) = L,     lim_x→a g(x) = M

If M ≠ 0, then the limit of the quotient is simply the quotients of the limits,

This is because for any infitesimal Δ x ≠0,

If L ≠ 0 and M = 0, then the limit

does not exist, because when Δx ≠ 0 is infinitesimal, f(a + Δx) has standard part L ≠ 0 and g(a + Δx) has standard part 0.

But what happens if both L and M are 0? In some cases a simple algebraic manipulation will enable us to compute the limit. For example,

even though both the numerator x² -1 and the denominator x + 1 approach 0 as x approaches -1.