limit as x approaches 0: 1-cosx/x
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To evaluate the limit as x approaches 0 of (1-cos(x))/x, we can manipulate the expression using trigonometric identities.
First, we can set up the fraction as (1 – cos(x))/(x * 1), where we multiply the denominator by 1 to maintain the same expression.
Next, we can use the trig identity that cos(x) = 1 – 2sin^2(x/2), which leads to:
1 – cos(x) = 1 – (1 – 2sin^2(x/2)) = 2sin^2(x/2)
Substituting this back into the original expression gives:
(2sin^2(x/2))/x
Now, we can use the limit law that states lim sin(x)/x = 1 as x approaches 0. This means that lim sin^2(x/2)/(x/2)^2 = 1, which implies lim sin^2(x/2)/((x/2) * (x/2)) = 1.
Substituting this back into our expression gives:
lim (2sin^2(x/2))/x = lim (2(x/2)^2 * (sin^2(x/2))/((x/2) * (x/2))) = lim ((sin(x/2))/(x/2))^2 * 2 = 2
Therefore, the limit as x approaches 0 of (1-cos(x))/x is equal to 2.
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