lim x->0 1-cosx/x
To find the limit of the function as x approaches 0, we can simplify the expression using some trigonometric identities
To find the limit of the function as x approaches 0, we can simplify the expression using some trigonometric identities.
First, we rewrite the expression (1 – cos(x))/x as (1/x) * (1 – cos(x)).
Next, we can use the trigonometric identity 1 – cos(x) = 2sin^2(x/2). So, our expression becomes:
(1/x) * 2sin^2(x/2).
Now, we can rewrite sin^2(x/2) as (sin(x/2))^2.
Our expression is now (1/x) * 2(sin(x/2))^2.
As x approaches 0, sin(x/2) also approaches 0. Therefore, (sin(x/2))^2 approaches 0.
Finally, since 1/x approaches ±infinity as x approaches 0, we conclude that the overall limit is ±infinity.
So, lim x->0 (1 – cos(x))/x = ±infinity, depending on the sign of 1/x.
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