Question

Lim(cos(4x)-cos^3(4x))/(3x^2) х стремится к 0

Answer 1

$\lim\limits _{x \to 0}\frac{cos4x-cos^34x}{3x^2}=\lim\limits _{x \to 0}\frac{cos4x\cdot (1-cos^24x)}{3x^2}=\lim\limits _{x \to 0}\frac{cos4x\cdot sin^24x}{3x^2}=\\\\=\Big [\; sina\sim a\; ,\; a\to 0\; \; \Rightarrow \; \; 4x\to 0\; ,\; sin4x\sim 4x\; \Big ]=\lim\limits _{x \to 0}\frac{cos4x\cdot (4x)^2}{3x^2}=\\\\=\lim\limits _{x \to 0}\frac{cos4x\cdot 16x^2}{3x^2}=\lim\limits _{x \to 0}\frac{cos4x\cdot 16}{3}=\Big [\; cos0=1\; \Big ]=\frac{1\cdot 16}{3}=\frac{16}{3}$

Answer 2

To find the limit of the expression as x approaches 0, let's analyze it step by step:

The expression is: (cos(4x) - cos^3(4x))/(3x^2)

As x approaches 0, we can make use of trigonometric identities to simplify the expression:

1. Using the trigonometric identity: cos(3θ) = 4cos^3(θ) - 3cos(θ) We can rewrite cos^3(4x) as (cos(4x))^3 = (cos(3 * 4x + x))^3 Applying the identity, we get: cos^3(4x) = (4cos(3 * 4x) - 3cos(4x))^3

2. Expanding the numerator: cos(4x) - cos^3(4x) = cos(4x) - (4cos(3 * 4x) - 3cos(4x))^3

3. Since we are looking for the limit as x approaches 0, we can use the Taylor series expansion: cos(x) = 1 - (x^2)/2 + (x^4)/24 + O(x^6) This expansion is valid for x approaching 0.

Now, let's apply the Taylor series expansion to the expression:

cos(4x) ≈ 1 - (4x)^2/2 + (4x)^4/24 = 1 - 8x^2 + 32x^4/24 = 1 - 8x^2 + 4x^4/3 cos(3 * 4x) = cos(12x) ≈ 1 - (12x)^2/2 + (12x)^4/24 = 1 - 72x^2 + 144x^4/24 = 1 - 72x^2 + 6x^4

Now, the numerator becomes:

cos(4x) - cos^3(4x) ≈ (1 - 8x^2 + 4x^4/3) - [(4 * (1 - 72x^2 + 6x^4)) - 3(1 - 8x^2 + 4x^4/3)]^3 ≈ (1 - 8x^2 + 4x^4/3) - [4 - 288x^2 + 24x^4 - 3 + 24x^2 - 4x^4/3]^3 ≈ (1 - 8x^2 + 4x^4/3) - (1 - 264x^2 + 20x^4)^3 ≈ 1 - 8x^2 + 4x^4/3 - (1 - 264x^2 + 20x^4)^3

Now we can divide this by (3x^2):

[(cos(4x) - cos^3(4x))/(3x^2)] ≈ [1 - 8x^2 + 4x^4/3 - (1 - 264x^2 + 20x^4)^3] / (3x^2)

As x approaches 0, all the terms with x^2 and x^4 become very small, and the expression will approach 0. Therefore, the limit as x approaches 0 of the given expression is 0.

Keep in mind that this is an approximation of the limit, and to get the exact limit, you'd need to perform more rigorous mathematical techniques or use software like L'Hôpital's rule or series expansion.