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

Answer 1

The limit is #2#.

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

#lim_(x ->0)(sin^2xcosx)/(1 - cosx)#
We will apply the identity #sin^2x+ cos^2x = 1 -> sin^2x = 1 - cos^2x#.
#=lim_(x ->0)((1 - cos^2x)cosx)/(1 - cosx)#
We can factor #1 - cos^2x# as a difference of squares
#=lim_(x->0) ((1 + cosx)(1 - cosx)(cosx))/(1 - cosx)#
#=lim_(x-> 0) ((1 + cosx)cancel(1 - cosx)(cosx))/(cancel(1- cosx))#
#=lim_(x-> 0) cosx +cos^2x#
#=cos^2(0) + cos(0)#
#=1 + 1#
#= 2#
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Answer 2

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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Answer from HIX Tutor

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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