Evaluate the Limit

Problem

(lim_x→0)((1−cos(x))/sin(x))

Solution

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

(1−cos(0))/sin(0)=(1−1)/0=0/0

2. Apply L'Hôpital's Rule, which states that if a limit results in 0/0 the limit is equal to the limit of the derivatives of the numerator and denominator.

(lim_x→0)(d(1−cos(x))/d(x)/d(sin(x))/d(x))

3. Differentiate the numerator and the denominator.

d(1−cos(x))/d(x)=sin(x)

d(sin(x))/d(x)=cos(x)

4. Substitute the derivatives back into the limit expression.

(lim_x→0)(sin(x)/cos(x))

5. Evaluate the limit by substituting x=0 into the new expression.

sin(0)/cos(0)=0/1=0

Final Answer

(lim_x→0)((1−cos(x))/sin(x))=0

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