Mikä on cos (3x) ^ (5 / x): n raja kuin x lähestyy 0?

Vastaus:

#lim_ (xto0) (cos (3x)) ^ (5 / x) = 1 #

Selitys:

# (Cos (3x)) ^ (5 / x) = e ^ (ln (cos (3x)) ^ (5 / x)) = e ^ ((5ln (cos (3x))) / x #

#lim_ (xto0) (5ln (cos (3x))) / x ## = 5lim_ (xto0) (ln (cos (3x))) / x = _ (DLH) ^ ((0/0)) #

# = 5lim_ (xto0) ((cos (3x)) (3x) ') / cos (3x) #

# = - 15lim_ (xto0) (sin (3x)) / cos (3x) #

# = _ (X-> 0, y> 0) ^ (3x = y) #

# -15lim_ (yto0) Siny / kodikas = lim_ (yto0) tany = 0 #

#lim_ (xto0) (cos (3x)) ^ (5 / x) = lim_ (xto0) e ^ ((5ln (cos (3x))) / x #

korvike

# (5ln (cos (3x))) / x = u #

# X-> 0 #

# U> 0 #

# = Lim_ (uto0) e ^ u = e ^ 0 = 1 #

kaavio {(cos (3x)) ^ (5 / x) -15.69, 16.35, -7.79, 8.22}