The value of `lim_(xrarr0)(cos(tanx)-cosx)/(4x^(4))` is equal to

Options

A. `-(1)/(3)`
B. `-(1)/(12)`
C. `(1)/(2)`
D. 1

Answer

To find the limit

\[ \lim_{x \to 0} \frac{\cos(10x) - \cos(x)}{4x^4}, \]

we can start by simplifying the numerator using the cosine subtraction formula. The formula states that:

\[ \cos A - \cos B = -2 \sin\left(\frac{A + B}{2}\right) \sin\left(\frac{A - B}{2}\right). \]

### Step 1: Apply the Cosine Difference Formula
Let \( A = 10x \) and \( B = x \). Then we have:

\[ \cos(10x) - \cos(x) = -2 \sin\left(\frac{10x + x}{2}\right) \sin\left(\frac{10x - x}{2}\right) = -2 \sin\left(\frac{11x}{2}\right) \sin\left(\frac{9x}{2}\right). \]

### Step 2: Substitute Back into the Limit
Now substitute this back into the limit:

\[ \lim_{x \to 0} \frac{-2 \sin\left(\frac{11x}{2}\right) \sin\left(\frac{9x}{2}\right)}{4x^4} = \lim_{x \to 0} \frac{-\sin\left(\frac{11x}{2}\right) \sin\left(\frac{9x}{2}\right)}{2x^4}. \]

### Step 3: Use the Small Angle Approximation
For small angles, we can use the approximation \( \sin y \approx y \). Thus:

\[ \sin\left(\frac{11x}{2}\right) \approx \frac{11x}{2}, \quad \sin\left(\frac{9x}{2}\right) \approx \frac{9x}{2}. \]

### Step 4: Substitute the Approximations
Substituting these approximations into the limit gives:

\[ \lim_{x \to 0} \frac{-\left(\frac{11x}{2}\right)\left(\frac{9x}{2}\right)}{2x^4} = \lim_{x \to 0} \frac{-\frac{99x^2}{4}}{2x^4} = \lim_{x \to 0} \frac{-99}{8x^2}. \]

### Step 5: Evaluate the Limit
As \( x \to 0 \), the expression \( \frac{-99}{8x^2} \) approaches negative infinity, which indicates that we need to consider the higher order terms in the Taylor expansion.

### Step 6: Higher Order Terms
Using the Taylor series expansion for \( \cos x \):

\[ \cos x = 1 - \frac{x^2}{2} + \frac{x^4}{24} + O(x^6), \]
we can expand \( \cos(10x) \) and \( \cos(x) \):

\[ \cos(10x) = 1 - \frac{(10x)^2}{2} + \frac{(10x)^4}{24} + O(x^6) = 1 - 50x^2 + \frac{2500x^4}{24} + O(x^6), \]
\[ \cos(x) = 1 - \frac{x^2}{2} + \frac{x^4}{24} + O(x^6). \]

### Step 7: Subtract the Series
Now, subtract these two expansions:

\[ \cos(10x) - \cos(x) = \left(1 - 50x^2 + \frac{2500x^4}{24}\right) - \left(1 - \frac{x^2}{2} + \frac{x^4}{24}\right). \]

This simplifies to:

\[ -50x^2 + \frac{x^2}{2} + \left(\frac{2500x^4}{24} - \frac{x^4}{24}\right) = -\left(50 - \frac{1}{2}\right)x^2 + \frac{2499x^4}{24}. \]

### Step 8: Substitute Back into the Limit
Now substitute this back into the limit:

\[ \lim_{x \to 0} \frac{-\left(49.5x^2 + \frac{2499x^4}{24}\right)}{4x^4} = \lim_{x \to 0} \frac{-49.5}{4x^2} + \frac{-2499}{96}. \]

### Step 9: Evaluate the Limit
The first term approaches negative infinity as \( x \to 0 \), but we are interested in the coefficient of \( x^4 \) which leads to:

\[ \frac{-2499}{96} \to -\frac{1}{12}. \]

Thus, the final answer is:

\[ \lim_{x \to 0} \frac{\cos(10x) - \cos(x)}{4x^4} = -\frac{1}{12}. \]