Tenentukan hasil dari limit f(x+h)-f(x)/h jika h mendekati 0 dan f(x)=sinx

Lim f(x + h) – f(x) / h
= Lim (sin (x + h) – sin x)/h
= Lim (sin x cos h + cos x sin h – sin x)/h
= Lim (sin x cos h – sin x + cos x sin h) / h
= Lim (sin x (cos h – 1) + cos x sin h) / h
= Lim (sin x . -2 sin^2 1/2 h + cos x sin h) / h
= Lim (-2 sin x . sin^2 1/2 h)/h + (cos x sin h)/h
= Lim -2 sin x sin 1/2 h . (sin 1/2 h)/h + cos x . (sin h)/h
= -2 sin x sin 1/2 (0) . 1/2 + cos x . 1
= -2 sin x . sin 0 . 1/2 + cos x
= 0 + cos x
= cos x

F(x) = sin x
f(x+h) = sin (x+h)

=> sin(x+h) – sinx
=> 2cos((x+h)+x)/2 ×sin((x+h)-x)/2
=> 2cos((2x+h)/2) × sin(h/2)

jadi

Lim [sin(x+h) – sin x]/h
Lim [2cos((2x+h)/2) × sin(h/2)]/h
Lim[2cos((2x+h)/2) × Limsin(h/2) /h

2cos((2x+0)/2) × 1/2
cos((2x)/2)
cosx

jwbannya
f'(x) = cos x