Evaluate the Limit

Problem

(lim_x→0)(tan2(3*x)/(2*x))

Solution

1. Identify the indeterminate form by substituting x=0 into the expression.

tan2(3*(0))/(2*(0))=0/0

2. Rewrite the expression to isolate the standard limit (lim_θ→0)(tan(θ)/θ)=1

tan2(3*x)/(2*x)=tan(3*x)/(3*x)⋅tan(3*x)⋅(3*x)/(2*x)

3. Simplify the constant terms and the variable x in the last fraction.

tan(3*x)/(3*x)⋅tan(3*x)⋅3/2

4. Apply the limit laws to each factor as x→0

(lim_x→0)(tan(3*x)/(3*x))=1

(lim_x→0)(tan(3*x))=0

(lim_x→0)(3/2)=3/2

5. Multiply the results of the individual limits together.

1⋅0⋅3/2=0

Final Answer

(lim_x→0)(tan2(3*x)/(2*x))=0

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