lim x to 1 of (sin(1-cosx))/x
Are you sure x is not approaching zero?
then you have
cos x = 1 - x^2/2 ......
and
sin (x^2/2)/x
= x/2
= 0
To find the limit of the given expression as x approaches 1, we can use the concept of L'Hôpital's rule. L'Hôpital's rule states that if the limit of the ratio of two functions is indeterminate (such as 0/0 or ∞/∞), the limit of the ratio of their derivatives will give the same result.
Let's apply L'Hôpital's rule to find the limit:
1. Take the derivative of the numerator and denominator separately:
numerator = sin(1 - cos(x))
denominator = x
Derivative of the numerator:
The derivative of sin(1 - cos(x)) can be found by applying the chain rule. The derivative of sin(u) is cos(u), and the derivative of the inside function 1 - cos(x) with respect to x is sin(x). So, we have cos(1 - cos(x)) * sin(x).
Derivative of the denominator:
The derivative of x with respect to x is 1.
2. Now, replace the original expression with the derivative ratio:
lim x → 1 (cos(1 - cos(x)) * sin(x)) / 1
3. Substitute x = 1 into the expression:
cos(1 - cos(1)) * sin(1) / 1 = cos(0) * sin(1) = 1 * sin(1) = sin(1)
Therefore, the limit of the given expression as x approaches 1 is sin(1).