How to Find the Limit of a Trigonometric Expression | Step-by-Step Guide

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

To find the limit of the given expression, we can use a trigonometric identity and apply a well-known limit

To find the limit of the given expression, we can use a trigonometric identity and apply a well-known limit. Let’s start by using the identity:

1 – cos(theta) = 2sin^2(theta/2)

Substituting this into the expression, we have:

lim theta->0 (2sin^2(theta/2)) / theta

Next, we can divide the numerator and denominator by theta/2:

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

Now, let’s use a known limit that as theta approaches 0, sin(theta)/theta approaches 1. Based on this, we can rewrite the expression as follows:

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

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

= lim theta->0 (8sin^2(theta/2)cos^2(theta/2)) / (theta/2) * (theta/2)

= lim theta->0 (8sin^2(theta/2)) / (theta/2) * (theta/2) * (cos^2(theta/2))

= lim theta->0 (8sin^2(theta/2)) / (theta/2) * (theta/2)

= lim theta->0 8(sin(theta/2)/ (theta/2))^2

= 8(1)^2

= 8

So, the limit of the expression as theta approaches 0 is 8.

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