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

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 apply algebraic manipulation

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

First, let’s rewrite the expression using the identity cos(0) = 1:

(1 – cosx)/x = (1 – cosx)/(x – 0)

Now, we can use the identity 2sin^2(x/2) = 1 – cosx:

(1 – cosx)/(x – 0) = (1 – cosx)/(2sin(x/2) * x/2)

Next, we can cancel out the common factor of cosx in the numerator and denominator:

(1 – cosx)/(2sin(x/2) * x/2) = 1/(2sin(x/2) * x/2)

Now, as x approaches 0, we can make the substitution x/2 = t:

1/(2sin(x/2) * x/2) = 1/(2sin(t) * t)

Finally, we can apply the limit:

lim x→0 (1 – cosx)/x = lim t→0 1/(2sin(t) * t)

To evaluate this limit, we can simplify the expression further by using the fact that sin(t)/t approaches 1 as t approaches 0:

lim t→0 1/(2sin(t) * t) = 1/(2 * 1 * 1) = 1/2

Therefore, the limit as x approaches 0 of (1 – cosx)/x is 1/2.

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