Solving the Limit | lim θ→0 (1 – cos(θ)) / θ using L’Hôpital’s Rule

lim theta->0 1 – cos(theta) / theta = ___________________

To solve the given limit:

lim θ→0 (1 – cos(θ)) / θ

Let’s use L’Hôpital’s Rule, which states that if the limit of a function f(x)/g(x) exists as x approaches a certain value, and f(x) and g(x) are both differentiable at that point, then the limit is equal to the derivative of f(x) divided by the derivative of g(x) evaluated at that point

To solve the given limit:

lim θ→0 (1 – cos(θ)) / θ

Let’s use L’Hôpital’s Rule, which states that if the limit of a function f(x)/g(x) exists as x approaches a certain value, and f(x) and g(x) are both differentiable at that point, then the limit is equal to the derivative of f(x) divided by the derivative of g(x) evaluated at that point.

Here, we have:

f(θ) = 1 – cos(θ)
g(θ) = θ

Taking the derivatives:

f'(θ) = 0 + sin(θ) (since the derivative of cos(θ) is -sin(θ))
g'(θ) = 1

Now, applying L’Hôpital’s Rule:

lim θ→0 (1 – cos(θ)) / θ = lim θ→0 sin(θ) / 1

As θ approaches 0, the limit becomes:

lim θ→0 (1 – cos(θ)) / θ = sin(0) / 1 = 0 / 1 = 0

Therefore, the value of the given limit is 0.

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