4)-
Find the following limit:
lim_(x->0) (1 - e^(2 x))/(sin(x))
Applying l'Hôpital's rule, we get that
lim_(x->0) (1 - e^(2 x))/(sin(x)) | = | lim_(x->0) ( d/( dx)(1 - e^(2 x)))/( d/( dx) sin(x))
| = | lim_(x->0) (-2 e^(2 x))/(cos(x))
| = | lim_(x->0) -(2 e^(2 x))/(cos(x))
lim_(x->0) -(2 e^(2 x))/(cos(x))
lim_(x->0) -(2 e^(2 x))/(cos(x)) = -(2 e^(2 0))/(cos(0)) = -2:
Answer: |-2
6)-
Find the following limit:
lim_(x->0) (sin^2(x))/(e^(5 x) - 1)
(sin^2(x))/(e^(5 x) - 1) = (sin^2(x))/(e^(5 x) - 1):
lim_(x->0) (sin^2(x))/(e^(5 x) - 1)
Applying l'Hôpital's rule, we get that
lim_(x->0) (sin^2(x))/(e^(5 x) - 1) | = | lim_(x->0) ( d/( dx) sin^2(x))/( d/( dx)(e^(5 x) - 1))
| = | lim_(x->0) (2 sin(x) cos(x))/(5 e^(5 x))
| = | lim_(x->0) 2/5 e^(-5 x) sin(x) cos(x)
lim_(x->0) 2/5 e^(-5 x) cos(x) sin(x)
lim_(x->0) 2/5 e^(-5 x) sin(x) cos(x) = 2/5 e^(-5 0) sin(0) cos(0) = 0:
Answer: | =0
