Finding the Limit of a Function as x Approaches 0 using Algebraic Simplification and Trigonometric Identities

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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