Question

Lim 1-cos6x/1-cos2xx стремиться к 0

Answer 1

Ответ:

9

Пошаговое объяснение:

По методу Лопиталя

Answer 2

To determine the limit of the expression (1 - cos(6x)) / (1 - cos(2x)) as x approaches 0, we can try to directly substitute x = 0 into the expression. However, this results in an indeterminate form (0/0), which means we need to perform further algebraic manipulation to simplify the expression and find its limit.

Let's start by using the trigonometric identity cos(2θ) = 1 - 2sin²θ to rewrite the denominator:

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1 - cos(2x) = 1 - (1 - 2sin²x) = 2sin²x

Now, the expression becomes:

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(1 - cos(6x)) / (2sin²x)

Using another trigonometric identity cos(2θ) = 2cos²θ - 1, we can express cos(6x) in terms of cos(2x):

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cos(6x) = 2cos²(3x) - 1

Now, substitute this back into the expression:

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(1 - (2cos²(3x) - 1)) / (2sin²x)

Simplify:

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(2 - 2cos²(3x)) / (2sin²x)

Now, factor out a 2 from the numerator:

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2(1 - cos²(3x)) / (2sin²x)

Apply the trigonometric identity 1 - cos²θ = sin²θ:

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2sin²(3x) / (2sin²x)

Cancel out the 2 and one of the sin²x terms:

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sin²(3x) / sin²x

Now, we have an expression that doesn't have an indeterminate form as x approaches 0. Substituting x = 0, we get:

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sin²(0) / sin²(0) = 0 / 0

We're still left with an indeterminate form. However, we can use the limit properties to evaluate this limit:

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lim(x->0) (sin²(3x) / sin²x) = lim(x->0) sin²(3x) * lim(x->0) (1 / sin²x)

As x approaches 0, sin²(3x) approaches 0 and (1 / sin²x) approaches infinity. This gives us an indeterminate form 0 * ∞.

In this case, to determine the limit, we should further analyze the behavior of sin²(3x) and (1 / sin²x) as x approaches 0. It seems that sin²(3x) is approaching 0 much faster than (1 / sin²x) is approaching infinity, resulting in an overall limit of 0.

However, this is a bit more intricate to prove formally, and it would involve using epsilon-delta definitions of limits or L'Hôpital's rule to handle the indeterminate form. If you are dealing with this in a calculus course or context, you might need to explore these concepts in more depth.