Evaluate the Limit

Problem

(lim_t→0)(tan(6*t)/sin(2*t))

Solution

1. Identify the indeterminate form by substituting t=0 into the expression, which results in tan(0)/sin(0)=0/0

2. Rewrite the expression using the fundamental trigonometric limit (lim_θ→0)(sin(θ)/θ)=1 and the definition tan(6*t)=sin(6*t)/cos(6*t)

3. Manipulate the expression to create the necessary forms for the limit laws.

tan(6*t)/sin(2*t)=sin(6*t)/(cos(6*t)*sin(2*t))

4. Multiply and divide by the appropriate factors of t to apply the limit properties.

sin(6*t)/(cos(6*t)*sin(2*t))=sin(6*t)/(6*t)⋅(2*t)/sin(2*t)⋅(6*t)/(2*t)⋅1/cos(6*t)

5. Simplify the constants and evaluate the limit of each part as t→0

(lim_t→0)(sin(6*t)/(6*t))=1

(lim_t→0)((2*t)/sin(2*t))=1

(lim_t→0)(1/cos(6*t))=1/1=1

6. Calculate the final result by multiplying the limits and the remaining constant.

1⋅1⋅6/2⋅1=3

Final Answer

(lim_t→0)(tan(6*t)/sin(2*t))=3

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