lim x->0 1-cosx/x
0
To evaluate this limit, we can use the following trigonometric identity:
lim x->0 (1 – cosx)/x = lim x->0 2sin^2(x/2)/x
We use this identity to rewrite the original expression as:
lim x->0 (1 – cosx)/x = lim x->0 2sin^2(x/2)/x
= lim x->0 2(x/2)^2 / (sin(x/2))(2)
= lim x->0 x/2 / sin(x/2)
Now, we can use the fact that sin(x) ~ x as x approaches 0 to simplify the expression further:
lim x->0 x/2 / sin(x/2) = lim x->0 1/2 / cos(x/2)
= 1/2
Therefore, the limit is 1/2.
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