Take the limit:
lim_(x->0) (1-sin(x))^(2/x)
Indeterminate form of type 1^infinity. Transform using lim_(x->0) (1-sin(x))^(2/x) = e^(lim_(x->0) (2 log(1-sin(x)))/x):
= e^(lim_(x->0) (2 log(1-sin(x)))/x)
Factor out constants:
= e^(2 (lim_(x->0) (log(1-sin(x)))/x))
Indeterminate form of type 0/0. Applying L'Hospital's rule we have, lim_(x->0) (log(1-sin(x)))/x = lim_(x->0) (( dlog(1-sin(x)))/( dx))/(( dx)/( dx)):
= e^(2 (lim_(x->0) (cos(x))/(-1+sin(x))))
The limit of a quotient is the quotient of the limits:
= exp((2 (lim_(x->0) cos(x)))/(lim_(x->0) (-1+sin(x))))
The limit of cos(x) as x approaches 0 is 1:
= e^(2/(lim_(x->0) (-1+sin(x))))
The limit of a constant is the constant:
The limit of a sum is the sum of the limits:
= e^(2/(-1+lim_(x->0) sin(x)))
The limit of sin(x) as x approaches 0 is 0:
Answer: |
| = 1/e^2
lim_(x->0) (1-sin(x))^(2/x)
Indeterminate form of type 1^infinity. Transform using lim_(x->0) (1-sin(x))^(2/x) = e^(lim_(x->0) (2 log(1-sin(x)))/x):
= e^(lim_(x->0) (2 log(1-sin(x)))/x)
Factor out constants:
= e^(2 (lim_(x->0) (log(1-sin(x)))/x))
Indeterminate form of type 0/0. Applying L'Hospital's rule we have, lim_(x->0) (log(1-sin(x)))/x = lim_(x->0) (( dlog(1-sin(x)))/( dx))/(( dx)/( dx)):
= e^(2 (lim_(x->0) (cos(x))/(-1+sin(x))))
The limit of a quotient is the quotient of the limits:
= exp((2 (lim_(x->0) cos(x)))/(lim_(x->0) (-1+sin(x))))
The limit of cos(x) as x approaches 0 is 1:
= e^(2/(lim_(x->0) (-1+sin(x))))
The limit of a constant is the constant:
The limit of a sum is the sum of the limits:
= e^(2/(-1+lim_(x->0) sin(x)))
The limit of sin(x) as x approaches 0 is 0:
Answer: |
| = 1/e^2
