Limit Calculation | (cos(x) – 1)/x as x approaches 0

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

To find the limit of the function f(x) = (cos(x) – 1)/x as x approaches 0, we can use algebraic manipulations and the properties of limits

To find the limit of the function f(x) = (cos(x) – 1)/x as x approaches 0, we can use algebraic manipulations and the properties of limits.

First, let’s simplify the given expression:
f(x) = (cos(x) – 1)/x

To eliminate the fraction, we can multiply both the numerator and denominator by the conjugate of the numerator, which is cos(x) + 1:
f(x) = (cos(x) – 1)/x * (cos(x) + 1)/(cos(x) + 1)
= (cos^2(x) – 1)/(x * (cos(x) + 1))

Now, we can apply the identity cos^2(x) – 1 = -sin^2(x):
f(x) = -sin^2(x)/(x * (cos(x) + 1))

We can further simplify by factoring out a negative sign:
f(x) = -(sin^2(x))/(x * (cos(x) + 1))

Now, let’s analyze the individual factors in the denominator as x approaches 0.

1. x:
As x approaches 0, the factor x goes to 0.

2. (cos(x) + 1):
As x approaches 0, cos(x) approaches cos(0) which is equal to 1. Therefore, the factor (cos(x) + 1) becomes 1 + 1 = 2.

3. sin^2(x):
As x approaches 0, sin(x) approaches sin(0) which is equal to 0. Therefore, the factor sin^2(x) becomes 0.

Putting all of this together, we have:
f(x) = -(sin^2(x))/(x * (cos(x) + 1))
= -(0)/(0 * 2)
= 0/0

At this point, we have obtained an indeterminate form of 0/0, which means we need to apply further techniques to evaluate the limit.

To proceed, we can use L’Hôpital’s Rule, which states that if a limit of the form 0/0 or ∞/∞ is encountered, we can take the derivative of both the numerator and denominator repeatedly until an answer is obtained.

Differentiating the numerator and denominator with respect to x:
f'(x) = -2sin(x)cos(x) – sin^2(x) / [(cos(x) + 1)^2 – x(2sin(x)cos(x))]

Now, let’s evaluate f'(x) as x approaches 0:
f'(0) = -2sin(0)cos(0) – sin^2(0) / [(cos(0) + 1)^2 – 0(2sin(0)cos(0))]
= 0 / (2^2 – 0)
= 0/4
= 0

Since we still get an indeterminate form of 0/0 after applying L’Hôpital’s Rule, we need to differentiate again:

Differentiating the numerator and denominator with respect to x:
f”(x) = -2cos^2(x) + 2sin^2(x) – 2sin^2(x)cos^2(x) – (-2sin(x)cos(x))(sin^2(x) + cos^2(x)) / [(cos(x) + 1)^2 – x(2sin(x)cos(x))]

Now, let’s evaluate f”(x) as x approaches 0:
f”(0) = -2cos^2(0) + 2sin^2(0) – 2sin^2(0)cos^2(0) – (-2sin(0)cos(0))(sin^2(0) + cos^2(0)) / [(cos(0) + 1)^2 – 0(2sin(0)cos(0))]
= -2 + 0 – 0 / (1 + 1)^2
= -2 / (2^2)
= -2/4
= -1/2

After taking the second derivative, we obtained a value of -1/2, which indicates the limit as x approaches 0. Hence, the final answer is:

lim x->0 (cos(x) – 1)/x = -1/2

More Answers: