Evaluate the Limit

Problem

(lim_x→0)(sin2(4*x)/(3*x))

Solution

1. Identify the limit form by substituting x=0 into the expression, which results in the indeterminate form 0/0

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

3. Factor out the constant 1/3 and split the squared sine term to manipulate the denominator.

1/3⋅(lim_x→0)((sin(4*x)⋅sin(4*x))/x)

4. Multiply and divide by 4 to create the required sin(4*x)/(4*x) structure.

4/3⋅(lim_x→0)(sin(4*x)/(4*x)⋅sin(4*x))

5. Apply the limit laws to evaluate the product of the two terms as x approaches 0

4/3⋅((lim_x→0)(sin(4*x)/(4*x)))⋅((lim_x→0)(sin(4*x)))

6. Substitute the known limit values, where (lim_x→0)(sin(4*x)/(4*x))=1 and (lim_x→0)(sin(4*x))=0

4/3⋅1⋅0=0

Final Answer

(lim_x→0)(sin2(4*x)/(3*x))=0

Want more problems? Check here!