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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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
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