lim x->0 1-cosx/x
To find the limit of the function as x approaches 0, we can use algebraic simplification and a trigonometric identity
To find the limit of the function as x approaches 0, we can use algebraic simplification and a trigonometric identity.
First, let’s rewrite the function 1 – cos(x)/x as (1 – cos(x))/x.
Next, we can simplify the numerator by using the trigonometric identity 2sin^2(x/2). This identity states that cos(x) = 1 – 2sin^2(x/2).
Substituting this identity into the numerator, we get:
(1 – (1 – 2sin^2(x/2)))/x
Simplifying further:
(1 – 1 + 2sin^2(x/2))/x
2sin^2(x/2)/x
Now, we can simplify the entire expression:
lim x->0 (2sin^2(x/2))/x
First, we notice that sin^2(x/2) and x are approaching 0 as x approaches 0. So, we can use the fact that sin(x)/x approaches 1 as x approaches 0.
lim x->0 (2sin^2(x/2))/x = 2 * lim x->0 (sin^2(x/2))/x
= 2 * lim x->0 (sin(x/2)/x) * sin(x/2)
Since sin(x/2)/x approaches 1 as x approaches 0 and sin(x/2) is always bounded, we have:
2 * 1 * sin(0)
= 2 * 1 * 0
= 0
Therefore, the limit of the function as x approaches 0 is 0.
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