Discovering the Limit of 1 – cos(x) / x as x Approaches 0 Using L’Hopital’s Rule

limit as x approaches 0: 1-cosx/x

0

To find the limit of the expression as x approaches 0, we can attempt to evaluate the expression directly. However, if we do so, we end up with a division by zero error:

1 – cos(0) / 0 = 1 – 1 / 0 = undefined

This means that the expression does not have a numerical value for x = 0. We will have to use a different approach to find the limit.

One way to find the limit is to use L’Hopital’s rule, which states that if we have a limit of the form 0/0 or infinity/infinity, we can take the derivative of the numerator and denominator separately and evaluate the limit of the resulting expression.

To apply L’Hopital’s rule, we will need to take the derivative of the numerator and denominator separately. The derivative of 1 is 0, and the derivative of cos(x) is -sin(x). Using the quotient rule, we get:

lim x->0 (1 – cos(x)) / x = lim x->0 (0 + sin(x)) / 1 = lim x->0 sin(x) / x = 1

Here, we used the fact that lim x->0 sin(x) / x = 1, which is a commonly known limit. Therefore, the original limit is equal to 1.

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

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