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, which states that if a limit results in 0/0 the limit of the ratio of the functions is equal to the limit of the ratio of their derivatives.

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

3. Differentiate the numerator and the denominator.

d(1−cos(x))/d(x)=sin(x)

d(x2)/d(x)=2*x

4. Rewrite the limit with the new expressions.

(lim_x→0)(sin(x)/(2*x))

5. Evaluate the new limit. Since substituting x=0 still results in 0/0 apply L'Hôpital's Rule a second time.

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

6. Differentiate again to find the final value.

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

(d(2)*x)/d(x)=2

7. Substitute x=0 into the resulting expression.

cos(0)/2=1/2

Final Answer

(lim_x→0)((1−cos(x))/(x2))=1/2

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