Evaluate the Limit

Problem

(lim_x→0)(sin(5*x)/(2*x))

Solution

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

2. Recall the fundamental trigonometric limit identity:

(lim_θ→0)(sin(θ)/θ)=1

3. Rewrite the expression to match the identity by factoring out the constant in the denominator:

1/2*(lim_x→0)(sin(5*x)/x)

4. Manipulate the expression by multiplying the numerator and denominator by 5 to make the argument of the sine function match the denominator:

5/2*(lim_x→0)(sin(5*x)/(5*x))

5. Apply the limit identity where θ=5*x As x→0 it follows that 5*x→0

5/2⋅1=5/2

Final Answer

(lim_x→0)(sin(5*x)/(2*x))=5/2

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