# How do you evaluate #(1-cos2x)/(x^2)# as x approaches 0?

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Use

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# = lim_(xrarr0)(2sin^2x)/x^2#

# = 2 (lim_(xrarr0)(sinx)/x)^2#

# = 2(1)^2 = 2#

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#### Explanation:

If we try to substitute 0 into

When direct substitution yields an indeterminate form, we can use L'Hôpital's rule:

Note that this is not the same as the Product Rule; rather, it means that the limit of the derivative of

Let's try this:

What happens when we substitute 0 back into this?

We end up with

When we substitute 0 into this, we get

This gives if

Thus, we know that

For more on L’Hôpital's rule, I encourage you to check out this link .

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