Evaluate lim->4 sin(2y)/tan(5y)
Here is what I have so far. I am not sure the next steps. Can someone help me?
1. sin(2y)/(sin(5y)*cos(5y))
2. (sin(2y)*cos(5y))/sin(5y)
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limit sin(2y)/tan(5y) as y->4
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To evaluate the limit lim→4 sin(2y)/tan(5y), let's simplify the expression step by step:
1. Start with the given expression: sin(2y)/tan(5y).
2. Since tan(θ) = sin(θ)/cos(θ), we can rewrite the expression in terms of sin and cos functions: sin(2y)/(sin(5y)/cos(5y)).
3. Now, let's simplify the expression using the rule of division: sin(2y) * cos(5y) / sin(5y).
4. Since we want to evaluate the limit as y approaches 4, substitute y = 4 into the expression: sin(2*4) * cos(5*4) / sin(5*4).
5. Compute the values inside the sine and cosine functions: sin(8) * cos(20) / sin(20).
6. Finally, evaluate the limit by directly plugging in the values: sin(8) * cos(20) / sin(20).
It is not possible to simplify this expression further without knowing the specific values of sin(8), cos(20), and sin(20). Therefore, the limit lim→4 sin(2y)/tan(5y) evaluates to sin(8) * cos(20) / sin(20).