Calculus, L'hopital's Rule

L'hopital's Rule

If f and g approach 0 as x approaches 0, and f′/g′ approaches l, then f/g approaches l.  This is known as L'hopital's rule, named after L'hopital (<biography>).  Note, some books call it L'Hospital's rule.

The result follows from the definition of f′ and g′.  The expression f(x)-f(0) over g(x)-g(0) is just f/g.  The same rule holds if x approaches any other real number, or infinity.  In the latter case, replace x with 1/x, and let x approach 0.  The rule is also applicable when both functions approach infinity; consider the reciprocal functions.

The rule may be invoked several times, until the fraction is no longer 0/0.  If f is 6x2+7 and g is 2x2-119, the limit of f/g as x approaches infinity is f′′/g′′ = 12/4 = 3.