lim as x approaches 0 of (3 sin 4x) / sin 3x
Lim x-->0 of (3 sin 4x) / sin 3x
Both the numerator and denominator are zero in this limit, so you can use L'Hopital's rule.
Lim x-->0 of (3 sin 4x) / sin 3x =
Lim x-->0 (12 cos 4x) / (3 cos 3x ) = 4
thanks Iblis
To find the limit as x approaches 0 of (3 sin 4x) / sin 3x, we can use the concept of L'Hôpital's Rule.
L'Hôpital's Rule states that if you have a limit of the form (f(x) / g(x)) as x approaches a particular value, and both f(x) and g(x) approach 0 or infinity (or some combination), then you can take the derivative of both f(x) and g(x) and evaluate the resulting limit.
In this case, both the numerator (3 sin 4x) and denominator (sin 3x) approach 0 as x approaches 0. So, we can apply L'Hôpital's Rule.
Let's differentiate the numerator and denominator individually:
The derivative of the numerator (3 sin 4x):
Using the chain rule, we differentiate sin 4x with respect to x, and multiply by the derivative of 4x, which is 4.
So, the derivative of 3 sin 4x is 12 cos 4x.
The derivative of the denominator (sin 3x):
Using the chain rule, we differentiate sin 3x with respect to x, and multiply by the derivative of 3x, which is 3.
So, the derivative of sin 3x is 3 cos 3x.
Now, we can rewrite the limit as x approaches 0 in terms of the derivatives:
lim as x approaches 0 of (3 sin 4x) / sin 3x = lim as x approaches 0 of (12 cos 4x) / (3 cos 3x)
Now, let's simplify further:
lim as x approaches 0 of (12 cos 4x) / (3 cos 3x) = (12/3) * (lim as x approaches 0 of cos 4x / cos 3x)
Simplifying the constant factor, we get:
lim as x approaches 0 of cos 4x / cos 3x = 4 * (lim as x approaches 0 of cos 4x / cos 3x)
Now, since we still have indeterminate form, we can apply L'Hôpital's Rule again.
Differentiating both the numerator and denominator:
The derivative of cos 4x is -4 sin 4x.
The derivative of cos 3x is -3 sin 3x.
We can rewrite the limit as x approaches 0 in terms of the derivatives once again:
4 * (lim as x approaches 0 of (-4 sin 4x) / (-3 sin 3x))
Simplifying further:
4 * (4/3) * (lim as x approaches 0 of sin 4x / sin 3x)
Now, when x approaches 0, sin 4x / sin 3x = (4x)/(3x) = 4/3.
So, the final result is:
lim as x approaches 0 of (3 sin 4x) / sin 3x = 4 * (4/3) = 16/3.
Therefore, the limit as x approaches 0 of (3 sin 4x) / sin 3x is 16/3.