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.