Evaluate the Limit

Problem

(lim_x→0)((3*(1−cos(x)))/x)

Solution

1. Identify the constant factor in the numerator and move it outside the limit using the scalar multiple property of limits.

(lim_x→0)((3*(1−cos(x)))/x)=3⋅(lim_x→0)((1−cos(x))/x)

2. Recognize the standard trigonometric limit identity which states that the limit of (1−cos(x))/x as x approaches 0 is equal to 0

(lim_x→0)((1−cos(x))/x)=0

3. Substitute the value of the standard limit back into the expression to find the final result.

3⋅0=0

Final Answer

(lim_x→0)((3*(1−cos(x)))/x)=0

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