L’Hôpital’s Rule: Finding the Limit as x Approaches 0 of (1 – cos(x))/x

limit as x approaches 0: 1-cosx/x

To find the limit as x approaches 0 for the given expression, we can apply L’Hôpital’s rule

To find the limit as x approaches 0 for the given expression, we can apply L’Hôpital’s rule.

L’Hôpital’s rule states that if the limit of f(x)/g(x) as x approaches a (where a can be a finite value or infinity) is an indeterminate form (such as 0/0 or infinity/infinity), then the limit of the derivative of f(x) divided by the derivative of g(x) is equivalent to the original limit.

In this case, we have the limit as x approaches 0 of (1 – cos(x))/x.

Since this expression evaluates to the indeterminate form 0/0 as x approaches 0, we can use L’Hôpital’s rule.

Taking the derivative of the numerator and the denominator, we have:

(1 – cos(x))’ = 0 – (-sin(x)) = sin(x)
x’ = 1

Now we can rewrite our limit using the derivatives:

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

As x approaches 0, sin(x) approaches 0. Therefore, the limit simplifies to:

lim(x→0) (sin(x))/1 = 0/1 = 0

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

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