If f(t) and g(t) are differentiable at t=a, and f(a)=g(a)=0, and either f'(a) or g'(a) is nonzero, then you can evaluate the limit of f(t)/g(t) as t->a (of the indeterminate form 0/0) by using linear approximation at the point t=a. It turns out that the limit equals f'(a)/g'(a) (which may be infinite if g'(a)=0 and f'(a)≠0). Geometrically, f'(a)/g'(a) is the slope of a parametric curve at the point (g(a),f(a))=(0,0), namely the curve traced out by coordinates (g(t),f(t)) as you vary the parameter (time) t. This provides some good intuition for l'Hospital's rule.