# How do you evaluate the limit #(sin^2xcosx)/(1-cosx)# as x approaches #0#?

The limit is

Let's start by simplifying the function because if we substitute directly we get an indeterminate form.

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To evaluate the limit of (sin^2xcosx)/(1-cosx) as x approaches 0, we can use algebraic manipulation and trigonometric identities. By factoring out sinx in the numerator and denominator, we get sinx(cosx)/(1-cosx).

Next, we can simplify further by using the identity sin^2x = 1 - cos^2x. Substituting this identity into the numerator, we have (1 - cos^2x)cosx.

Now, we can cancel out the common factor of cosx in the numerator and denominator, resulting in (1 - cos^2x)/(1 - cosx).

Since cos^2x and cosx both approach 1 as x approaches 0, we can substitute 1 for both of them in the expression.

Thus, the limit becomes (1 - 1)/(1 - 1) = 0/0.

This is an indeterminate form, so we need to further simplify. By factoring out a common factor of (1 - cosx) in the numerator, we get (1 - cosx)(1 + cosx)/(1 - cosx).

Now, we can cancel out the common factor of (1 - cosx), resulting in (1 + cosx).

Finally, as x approaches 0, cosx approaches 1, so the limit evaluates to 1 + 1 = 2.

Therefore, the limit of (sin^2xcosx)/(1-cosx) as x approaches 0 is 2.

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