lim as x-> 0 of 3x/(sin2x)
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lim as x-> 0 of 3x/(sin2x)
3 * lim as x-> 0 of [ x/(sin2x) ]
as x--0, sin 2x --> 2x - (2x)^3/3! etc which approaches 2x when x is small
3 * x/(2x) = 3/2
To find the limit as x approaches 0 of (3x)/(sin(2x)), we can use L'Hôpital's Rule.
L'Hôpital's Rule states that if we have a limit of the form "0/0" or "∞/∞", we can take the derivative of the numerator and denominator separately and then evaluate the limit again.
Let's apply L'Hôpital's Rule to our expression:
Take the derivative of the numerator:
d/dx (3x) = 3
Take the derivative of the denominator:
d/dx (sin(2x)) = 2cos(2x)
Now, let's evaluate the limit again using the new derivatives:
lim (x->0) (3)/(2cos(2x))
Plugging in x = 0 into the expression, we get:
lim (x->0) (3)/(2cos(0))
lim (x->0) (3)/(2*1)
lim (x->0) 3/2
Therefore, the limit as x approaches 0 of (3x)/(sin(2x)) is 3/2.