f’(x) = lim x → h of (f(x+h) - f(x)) / h and g’(x) = lim h → 0 of (g(x+h) - g(x)) / h.

So f’(x)/g’(x) = (lim h → 0 of (f(x+h) - f(x)) / h) / (lim h → 0 of (g(x+h) - g(x)) / h)

= lim h → 0 of (f(x+h) - f(x)) / (g(x+h) - g(x))

So lim x → a of f’(x)/g’(x) = lim x → a lim h → 0 of (f(x+h) - f(x)) / (g(x+h) - g(x))

Plug in x = a and the - f(x) and - g(x) terms disappear, since we’re given f(a) = g(a) = 0. Then un-plug-in x = a to keep the rest of the limit.

lim x → a of f’(x) / g’(x) = lim x → a lim h → 0 of f(x+h) / g(x+h)

Plug in h = 0 limit

lim x → a of f’(x) / g’(x) = lim x → a of f(x) / g(x), QED ¯\_(ツ)_/¯