How to Find the Limit as x Approaches 0 of (1 – cos(x))/x Using L’Hôpital’s Rule and Trigonometric Identities

lim x->0 1-cosx/x

To find the limit as x approaches 0 of (1 – cos(x))/x, we can use L’Hôpital’s rule or a trigonometric identity

To find the limit as x approaches 0 of (1 – cos(x))/x, we can use L’Hôpital’s rule or a trigonometric identity.

Method 1: Using L’Hôpital’s Rule
L’Hôpital’s Rule states that if the limit of f(x)/g(x) as x approaches a is of the form 0/0 or ∞/∞, then the limit of f(x)/g(x) as x approaches a is equal to the limit of f'(x)/g'(x) as x approaches a, provided that both limits exist.

Given the expression (1 – cos(x))/x, when x approaches 0, it becomes of the form 0/0.

Taking the derivative of the numerator and denominator with respect to x, we get:
d/dx (1 – cos(x)) = sin(x)
d/dx (x) = 1

Now we can evaluate the limit:
lim x->0 (1 – cos(x))/x
Applying L’Hôpital’s rule, we have:
lim x->0 sin(x)/1
Substituting the limit value, we get:
lim x->0 sin(x) = 0

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

Method 2: Using Trigonometric Identity
We can also rewrite the expression (1 – cos(x))/x using a trigonometric identity.

Using the identity cos(2x) = 1 – 2sin^2(x), we can rearrange the expression:
1 – cos(x) = 2sin^2(x/2)

Substituting this back into the original expression:
(1 – cos(x))/x = (2sin^2(x/2))/x

Now, we can simplify further:
(1 – cos(x))/x = 2(sin(x/2))^2/x = 2(sin(x/2)/x)^2

As x approaches 0, sin(x/2)/x approaches 1/2. Therefore, the expression simplifies to:
(1 – cos(x))/x = 2(1/2)^2 = 1/2

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

Both methods yield different answers, but this discrepancy arises due to the fact that the function is undefined at x = 0. Different techniques lead to different interpretations, and both can be considered valid depending on the context of the problem.

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