Finding the Limit as x Approaches 0 of (1-cosx)/x: Understanding Trigonometric Simplification

limit as x approaches 0: 1-cosx/x

To find the limit as x approaches 0 of the expression (1-cosx)/x, we can simplify the expression by using a trigonometric identity

To find the limit as x approaches 0 of the expression (1-cosx)/x, we can simplify the expression by using a trigonometric identity.

The identity we can use here is 1 – cosx = 2sin^2(x/2). Rewriting the expression, we have:

(1 – cosx)/x = (2sin^2(x/2))/x.

Now we can simplify further by factoring out sin(x/2) in the numerator and cancelling it with x in the denominator:

(2sin^2(x/2))/x = 2(sin(x/2)/x) * sin(x/2).

As x approaches 0, sin(x/2)/x also approaches 0 since sin(x/2) approaches 0 faster than x approaches 0. Therefore, the entire expression (1-cosx)/x approaches 2(0) = 0 as x approaches 0.

Therefore, the limit as x approaches 0 of (1-cosx)/x is 0.

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