Finding the Limit of (1-cos(x))/x as x Approaches 0 Using L’Hospital’s Rule

lim x->0 (1-cosx)/x

To find the limit of (1-cos(x))/x as x approaches 0, we can use L’Hospital’s Rule

To find the limit of (1-cos(x))/x as x approaches 0, we can use L’Hospital’s Rule.

L’Hospital’s Rule states that for an indeterminate form of type 0/0 or infinity/infinity, if the derivative of the numerator and the derivative of the denominator both exist and the limit of their quotient exists, then the limit of the original function also exists and is equal to the limit of the quotient of their derivatives.

Let’s apply L’Hospital’s Rule to the limit (1-cos(x))/x as x approaches 0:

lim x->0 (1-cos(x))/x

Taking the derivatives of the numerator and the denominator:
Numerator: d(1-cos(x))/dx = 0 + sin(x) = sin(x)
Denominator: d(x)/dx = 1

Now, we can evaluate the limit of the quotient of their derivatives:

lim x->0 (sin(x))/1

Since sin(x) approaches 0 as x approaches 0, the limit becomes:

lim x->0 0/1 = 0

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

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