limit as x approaches 0: 1-cosx/x
0
To find the limit as x approaches 0 of 1-cosx/x, we can use the L’Hospital’s rule or trigonometric identities.
Using L’Hospital’s rule:
lim x→0 (1-cosx)/x
= lim x→0 sinx/x (since 1-cosx can be simplified to 2sin^2(x/2))
= 1 (by the definition of the limit)
Therefore, the limit of 1-cosx/x as x approaches 0 is equal to 1.
Using trigonometric identity:
We can use the identity lim x→0 sinx/x = 1 to simplify the expression:
1-cosx/x = (1-cosx)/(x/2) * 2
= 2sin^2(x/2) / x * 2
= (sin(x/2)/(x/2))^2 * 1
= 1 * 1
= 1
Therefore, the limit of 1-cosx/x as x approaches 0 is equal to 1.
Both methods give the same answer, which is that the limit of 1-cosx/x as x approaches 0 is equal to 1.
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