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

Problem

(lim_x→0)((cos(x)−1)/x)

Solution

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

2. Evaluate the numerator and denominator at the limit point to find (cos(0)−1)/0=(1−1)/0=0/0 which is an indeterminate form.

3. 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.

4. Differentiate the numerator cos(x)−1 to get −sin(x) and the denominator x to get 1

5. Rewrite the limit using these derivatives.

(lim_x→0)((−sin(x))/1)

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

(−sin(0))/1=0/1=0

Final Answer

(lim_x→0)((cos(x)−1)/x)=0

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