lim x->0 (cosx-1)/x
0
To evaluate this limit, we can simplify the expression using trigonometric identities. Recall that cos(0) = 1, so substituting x = 0 directly would result in division by zero. Instead, we can manipulate the expression as follows:
(cosx – 1)/x = (cosx – 1)(cosx + 1)/(x(cosx + 1)) = -sin^2(x)/(x(cosx + 1))
Now, as x approaches 0, sin(x)/x approaches 1. We can use this fact and multiply the expression by sin(x)/sin(x):
-lim x->0 (sin(x)/x) * (sin(x))/(cos(x) + 1)
We know that the limit of sin(x)/x as x approaches 0 is equal to 1, so we can substitute 1 in for sin(x)/x. Then we get:
-lim x->0 (sin(x)/(cos(x) + 1)) = – sin(0)/(cos(0) + 1) = 0.
Therefore, the limit of (cosx – 1)/x as x approaches 0 is equal to 0.
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