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.