Evaluate the Limit

Problem

(lim_x→0)((sin(3*x)*sin(5*x))/(x2))

Solution

1. Rewrite the expression to group the trigonometric functions with their corresponding variables in the denominator.

(lim_x→0)(sin(3*x)/x⋅sin(5*x)/x)

2. Apply the product rule for limits, which states that the limit of a product is the product of the limits.

((lim_x→0)(sin(3*x)/x))⋅((lim_x→0)(sin(5*x)/x))

3. Adjust the denominators to match the arguments of the sine functions by multiplying and dividing by the necessary constants.

((lim_x→0)((3*sin(3*x))/(3*x)))⋅((lim_x→0)((5*sin(5*x))/(5*x)))

4. Factor out the constants from the limits.

3*((lim_x→0)(sin(3*x)/(3*x)))⋅5*((lim_x→0)(sin(5*x)/(5*x)))

5. Use the fundamental trigonometric limit identity (lim_θ→0)(sin(θ)/θ)=1

3*(1)⋅5*(1)

6. Multiply the resulting values to find the final limit.

15

Final Answer

(lim_x→0)((sin(3*x)*sin(5*x))/(x2))=15

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