Evaluating The Limit Of (1-Cos(Theta)) / Theta: L’Hopital’S Rule And Trigonometric Identities

lim theta->0 1 – cos(theta) / theta = ___________________

0

To evaluate the limit:

lim theta->0 (1 – cos(theta)) / theta

We can use L’Hopital’s rule or trigonometric identities:

Method 1: L’Hopital’s Rule

Since we have the indeterminate form 0/0, we can use L’Hopital’s rule to evaluate the limit by taking the derivative of the numerator and denominator with respect to theta:

lim theta->0 (1 – cos(theta)) / theta

= lim theta->0 sin(theta) / 1 (using L’Hopital’s rule)

= sin(0) / 1

= 0 / 1

= 0

Therefore, the limit is 0.

Method 2: Trigonometric Identities

We can rewrite the expression as:

lim theta->0 (1 – cos(theta)) /theta * (1 + cos(theta)) / (1 + cos(theta))

= lim theta->0 (1 – cos^2(theta)) / (theta * (1 + cos(theta)))

= lim theta->0 sin^2(theta) / (theta * (1 + cos(theta))) (using the trig identity: sin^2(theta) = 1 – cos^2(theta))

= lim theta->0 sin(theta) / (1 + cos(theta)) * sin(theta) / theta

= 1/2 * 1/1 * 0/0 (using the trig identity: sin(2theta) = 2sin(theta)cos(theta))

= 0

Therefore, the limit is 0.

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