Calculus limits ? why is the limit of (tan7x) / (sin3x) as x approaches 0. Is 7/3 ?
Calculus limits ? why is the limit of (tan7x) / (sin3x) as x approaches 0. Is 7/3 ? why is the limit of (tan7x) / (sin3x) as x approaches 0. Is 7/3 ?
The solution is (7 / (3cos7x)) * (sin7x / 7x) * (3x / sin3x ) = 7/3 I just don't understand why we have (7 / (3cos7x)) in the process ?? PLEASE HELP, I am quite confused ! thank you in advance !
= limx->0 7/3 [tan7x/7x * 3x/sin3x]
= 7/3[ limx->0 tan7x/7x *limx--/.3x/sin3x ]
= 7/x (1 * 1)
that'd be the way i solve it
The solution is (7 / (3cos7x)) * (sin7x / 7x) * (3x / sin3x )
u ed up with
limx-> 0 = tan7x/sin3x
= lim-> 0 7/(3c0s7x)
note that in that step there still is that lim-> 0 thingy so u have to put x= 0
= lim-> 0 7/(3c0s7x)
bcoz cos0 = 1
I have no idea what you're trying to do.
I would use l'HÃ´pital's rule (link below)
limit(x -> 0) [tan(7x) / sin(3x)]
= limit(x -> 0) [7sec^2(x) / 3cos(3x)]
= 7(1) / 3(1)
I think an easier way to do this is using l/hopitals rule, I'm not sure about your method, take the derivative of the top, and devide by the derivative of the bottom, until there is a determinate answer, in this case it is only (this only works in cases where there is 0/0 or infinity/infinity. first derivative of the top is 7 * secant ^2(7x), first derivative of bottom is 3 cos (3x) evaluate at 0 and you get 7/3
You need to know that tanA = sinA/cosA is an identity and that lim(sinA/A) = 1 as A approaches 0. From these you get:
tan7x = sin7x / cos7x is an identity, and lim(sin7x / 7x) = 1 and lim(3x/sin3x) = 1 as x approaches 0.
Put these facts together:
(tan7x) / (sin3x) = [(sin7x) / (cos7x ) ] / sin3x
= [(sin7x / 7x) (7x / cos7x)] (1/sin3x)
= (7x / 3x) (sin7x / 7x) (1/cos7x) (3x / sin3x)
= (7/3) (sin7x / 7x) (1/cos7x) (3x / sin3x), provided x is not zero.
Now take limit as x approaches zero and get:
(7/3) lim(sin7x / 7x) lim(1/cos7x) lim(3x / sin3x)
= (7/3) (1) (1/cos0)(1)
= (7/3) (1) (1) (1)
According t l'Hospital',s rule, The limit of a quotient is the limit of the quotient of the derivatives taken separately. Also, the derivative of a composite function is the derivative of the first function times the derivative of the interior function.
The derivative of tan7x= 7sec^2x and secant = 1/ cos
The derivative of sin3x is 3 cos x
The quotient of the derivatives is (7/3)*1/cos^3x
The limit of cosx as x approaches 0 is 1.
(7/3)*1 is 7/3
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