Evaluating the Limit as x Approaches 0 of (1 – cosx)/x using Trigonometric Identity and Algebraic Manipulations

limit as x approaches 0: 1-cosx/x

To find the limit as x approaches 0 of the expression (1 – cosx)/x, we can use a trigonometric identity and algebraic manipulations

To find the limit as x approaches 0 of the expression (1 – cosx)/x, we can use a trigonometric identity and algebraic manipulations.

First, let’s rewrite the expression using the double-angle formula for cosine:
1 – cosx = 2sin^2(x/2)

Now, the expression becomes:
(2sin^2(x/2))/x

Next, let’s simplify by factoring out a sin(x/2) term:
(2sin(x/2) * sin(x/2))/x

Both sin(x/2) terms tend to 0 as x approaches 0, so the limit can be determined by evaluating:
lim(x->0) 2sin(x/2)/x

We can further simplify by using the Squeeze Theorem or by recognizing that sin(x)/x has a well-known limit of 1 as x approaches 0:
lim(x->0) 2 * (sin(x/2)/(x/2)) * (1/2)

The expression now becomes:
2 * 1 * (1/2) = 1

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

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