lim x -> ∞ (x * sin(4/x) - x * tan(4/x))/(cos(4/x) - 1) =***

To evaluate the limit

\[
\lim_{x \to \infty} \frac{x \cdot \sin\left(\frac{4}{x}\right) - x \cdot \tan\left(\frac{4}{x}\right)}{\cos\left(\frac{4}{x}\right) - 1},
\]

we can follow these steps:

### Step 1: Simplify the expressions as \(x \to \infty\)
As \(x \to \infty\), \(\frac{4}{x} \to 0\). So we can use the small angle approximations for sine, cosine, and tangent:
\[
\sin\left(\frac{4}{x}\right) \approx \frac{4}{x}, \quad \tan\left(\frac{4}{x}\right) \approx \frac{4}{x}, \quad \cos\left(\frac{4}{x}\right) \approx 1 - \frac{1}{2}\left(\frac{4}{x}\right)^2
\]

### Step 2: Substitute the approximations into the limit
Substituting these into the limit expression gives:
\[
\frac{x \cdot \frac{4}{x} - x \cdot \frac{4}{x}}{\left(1 - \frac{1}{2}\left(\frac{4}{x}\right)^2\right) - 1}
\]

Simplifying the numerator:
\[
x \cdot \frac{4}{x} - x \cdot \frac{4}{x} = 4 - 4 = 0
\]

The denominator simplifies to:
\[
1 - 1 + \frac{8}{x^2} = \frac{8}{x^2

Learn how to evaluate the limit lim x -> ∞ (x * sin(4/x) - x * tan(4/x))/(cos(4/x) - 1) with a detailed step-by-step solution. This problem involves advanced trigonometric functions and limit evaluation techniques suitable for advanced college-level mathematics.

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