Evaluating (1 – Cos(Theta)) / Theta Using Trigonometric Limits And L’Hopital’S Rule

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 the trigonometric limit:

lim theta->0 sin(theta) / theta = 1

First, we can simplify the expression by multiplying both numerator and denominator by (1+cos(theta)):

(1-cos(theta)) / theta * (1+cos(theta)) / (1+cos(theta))

= (1 – cos^2(theta)) / (theta * (1 + cos(theta)))

= sin^2(theta) / (theta * (1 + cos(theta)))

Now, we can substitute sin(theta) / theta = 1:

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

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

= 1^2 / (1+1)

= 1/2

Therefore, the limit as theta approaches 0 of (1 – cos(theta)) / theta is 1/2.

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