Evaluate the Limit ( limit as x approaches 0 of 1-cos(x))/x

Problem

(lim_x→0)((1−cos(x))/x)

Solution

1. Identify the form of the limit by substituting x=0 into the expression.

2. Evaluate the numerator and denominator at x=0 which results in (1−cos(0))/0=(1−1)/0=0/0

3. Apply L'Hôpital's Rule because the limit is in the indeterminate form 0/0

4. Differentiate the numerator and the denominator with respect to x separately.

5. Calculate the derivative of the numerator: d(1−cos(x))/d(x)=sin(x)

6. Calculate the derivative of the denominator: d(x)/d(x)=1

7. Rewrite the limit using these derivatives: (lim_x→0)(sin(x)/1)

8. Substitute x=0 into the new expression to find the limit value.

9. Simplify the result: sin(0)/1=0/1=0

Final Answer

(lim_x→0)((1−cos(x))/x)=0

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