How do you find the Limit of #[(sin x) * (sin^2 x)] / [1 -( cos x)]# as x approaches 0?

Answer 1

Perform some conjugate multiplication and simplify to get #lim_(x->0)(sinx*sin^2x)/(1-cosx)=0#

Direct substitution produces indeterminate form #0/0#, so we'll have to try something else.
Try multiplying #(sinx*sin^2x)/(1-cosx)# by #(1+cosx)/(1+cosx)#: #(sinx*sin^2x)/(1-cosx)*(1+cosx)/(1+cosx)#
#=(sinx*sin^2x(1+cosx))/((1-cosx)(1+cosx))#
#=(sinx*sin^2x(1+cosx))/(1-cos^2x)#
This technique is known as conjugate multiplication, and it works nearly every time. The idea is to use the difference of squares property #(a-b)(a+b)=a^2-b^2# to simplify either the numerator or denominator (in this case the denominator).
Recall that #sin^2x+cos^2x=1#, or #sin^2x=1-cos^2x#. We can therefore replace the denominator, which is #1-cos^2x#, with #sin^2x#: #((sinx)(sin^2x)(1+cosx))/(sin^2x)#
Now the #sin^2x# cancels: #((sinx)(cancel(sin^2x))(1+cosx))/(cancel(sin^2x))# #=(sinx)(1+cosx)#
Finish by taking the limit of this expression: #lim_(x->0)(sinx)(1+cosx)# #=lim_(x->0)(sinx)lim_(x->0)(1+cosx)# #=(0)(2)# #=0#
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Answer 2

To find the limit of the given expression as x approaches 0, we can use algebraic manipulation and trigonometric identities.

First, let's simplify the expression:

[(sin x) * (sin^2 x)] / [1 - (cos x)]

Using the identity sin^2(x) = 1 - cos^2(x), we can rewrite the expression as:

[(sin x) * (1 - cos^2 x)] / [1 - cos x]

Next, we can factor out sin x from the numerator:

[sin x * (1 - cos x) * (1 + cos x)] / [1 - cos x]

Now, we can cancel out the common factor of (1 - cos x) in the numerator and denominator:

[sin x * (1 + cos x)] / 1

Finally, as x approaches 0, sin x approaches 0 and cos x approaches 1. Therefore, the limit of the expression is:

0 * (1 + 1) / 1 = 0.

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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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