Evaluate the Limit

Problem

(lim_x→0)((1−cos(x))/(x2))

Solution

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

(1−cos(0))/0=(1−1)/0=0/0

2. Apply L'Hôpital's Rule by differentiating the numerator and the denominator with respect to x

(lim_x→0)(d(1−cos(x))/d(x)/d(x2)/d(x))

3. Calculate the derivatives of the numerator and denominator.

(lim_x→0)(sin(x)/(2*x))

4. Identify the indeterminate form again as x approaches 0

sin(0)/(2*(0))=0/0

5. Apply L'Hôpital's Rule a second time.

(lim_x→0)(d(sin(x))/d(x)/(d(2)*x)/d(x))

6. Calculate the derivatives and evaluate the limit.

(lim_x→0)(cos(x)/2)

7. Substitute x=0 into the simplified expression.

cos(0)/2=1/2

Final Answer

(lim_x→0)((1−cos(x))/(x2))=1/2

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