Evaluate the Limit limit as x approaches 0 of (sin(x))/(5x)

Problem

(lim_x→0)(sin(x)/(5*x))

Solution

1. Identify the constant factor in the denominator. The expression can be rewritten by pulling out the constant 1/5 from the limit.

(lim_x→0)(1/5)⋅sin(x)/x

2. Apply the constant multiple rule for limits, which allows the constant to be moved outside the limit operation.

1/5⋅(lim_x→0)(sin(x)/x)

3. Recall the fundamental trigonometric limit identity, which states that the limit of sin(x)/x as x approaches 0 is equal to 1

(lim_x→0)(sin(x)/x)=1

4. Substitute the value of the fundamental limit back into the expression to find the final result.

1/5⋅1=1/5

Final Answer

(lim_x→0)(sin(x)/(5*x))=1/5

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