How To Evaluate The Limit Using L’Hopital’S Rule And Small-Angle Approximation In Mathematics

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

0

To evaluate this limit, we can use L’Hopital’s rule or the small-angle approximation.

Using L’Hopital’s rule:
We differentiate both the numerator and denominator with respect to theta, which gives us:

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

Using the small-angle approximation:
We can approximate cos(theta) as 1 and sin(theta) as approximately equal to theta for small values of theta. Thus, we can rewrite the expression as:

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

Therefore, the limit is equal to 0.

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