מהי הגבלה של cos (3x) ^ (5 / x) כאשר x מתקרב 0?

תשובה:

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

הסבר:

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

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

# (5xlim) (xto0) (cos (3x)) '(3x)') / cos (3x) #

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

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

#-15lim_ (yto0) siny / cozy = lim_ (yto0) tany = 0 #

# (xo0) (cos (3x)) (5 / x) = lim_ (xto0) e ^ ((5ln (cos (3x)) / x #

תחליף

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

# x-> 0 #

# u-> 0 #

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

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