Solving the Limit of (1 – cos(x))/x as x approaches 0 using Trigonometric Manipulation

limit as x approaches 0: 1-cosx/x

(1 – cos(x))/x is 2

To find the limit of the function f(x) = (1 – cos(x))/x as x approaches 0, we can use algebraic manipulation and the concept of a trigonometric limit.

Step 1: Simplify the expression
Start by applying the trigonometric identity: cos(x) = 1 – 2sin^2(x/2). This identity will help simplify the expression.
So, 1 – cos(x) becomes 1 – (1 – 2sin^2(x/2)) = 2sin^2(x/2).

The expression now becomes: f(x) = (2sin^2(x/2))/x.

Step 2: Apply the trigonometric limit
We know that as x approaches 0, sin(x/2)/(x/2) approaches 1. This is a commonly known trigonometric limit. Therefore, we can replace sin(x/2)/(x/2) with 1 in our expression.

The expression now becomes: f(x) = 2(1)^2 = 2.

Step 3: Determine the limit
Now, as x approaches 0, f(x) approaches 2.

Hence, the limit as x approaches 0 for the function (1 – cos(x))/x is 2.

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