How do you find the limit of #(sin4x)/(sin6x)# as #x->0#?

Answer 1

#2/3#

Multiply and divide by #4x# and by #6x#, then rearrange as follows:
#sin(4x)/sin(6x) * (4x)/(4x) * (6x)/(6x) = sin(4x)/(4x) * (6x)/sin(6x) * (4x)/(6x)#

As you might have noticed, now we have in the first two terms the known limit

#lim_{y\to 0} sin(y)/y#
which is always #1#, no matter what expression we insert as #y#, as long as it tends to zero (and of course, both #4x# and #6x# tend to zero as #x\to 0#).
The third term is simply #4/6=2/3#, since we can cancel the #x# factor.
So, the product of the limits is #1*1*2/3=2/3#
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Answer 2

To find the limit of (sin4x)/(sin6x) as x approaches 0, we can use the concept of limits and trigonometric identities. By applying the limit properties and the fact that sin(x)/x approaches 1 as x approaches 0, we can simplify the expression.

Using the trigonometric identity sin(2θ) = 2sin(θ)cos(θ), we can rewrite the expression as (2sin(2x)cos(2x))/(2sin(3x)cos(3x)).

Next, we can cancel out the common factors of 2 and sin(2x) to get cos(2x)/(sin(3x)cos(3x)).

Further simplifying, we can rewrite cos(2x) as 1 - 2sin^2(x) and cos(3x) as 1 - 3sin^2(x).

Substituting these values, we get (1 - 2sin^2(x))/(sin(3x)(1 - 3sin^2(x))).

Now, as x approaches 0, sin(x) also approaches 0. Therefore, we can substitute sin(x) with 0 in the expression.

Doing so, we get (1 - 2(0))/(0(1 - 3(0))).

Simplifying further, we have 1/0.

Since division by zero is undefined, the limit of (sin4x)/(sin6x) as x approaches 0 does not exist.

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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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