lim x -> ∞ (2x ^ 2 * tan(1/x) - x * sin(1/x) + 1/x)/(x * cos(2/x)) = A. 2 B. 1 C. D. -1 E.-2

We need to evaluate the limit of the expression as \( x \to \infty \):

\[
\lim_{x \to \infty} \frac{2x^2 \cdot \tan\left(\frac{1}{x}\right) - x \cdot \sin\left(\frac{1}{x}\right) + \frac{1}{x}}{x \cdot \cos\left(\frac{2}{x}\right)}
\]

Let's break this down step by step:

### Step 1: Analyze small-angle approximations
As \( x \to \infty \), the terms \( \frac{1}{x} \) and \( \frac{2}{x} \) become small, so we can apply small-angle approximations for trigonometric functions:

- For \( \tan\left(\frac{1}{x}\right) \), use \( \tan z \approx z \) for small \( z \), so:
  \[
  \tan\left(\frac{1}{x}\right) \approx \frac{1}{x}.
  \]

- For \( \sin\left(\frac{1}{x}\right) \), use \( \sin z \approx z \) for small \( z \), so:
  \[
  \sin\left(\frac{1}{x}\right) \approx \frac{1}{x}.
  \]

- For \( \cos\left(\frac{2}{x}\right) \), use \( \cos z \approx 1 \) for small \( z \), so:
  \[
  \cos\left(\frac{2}{x}\right) \approx 1.
  \]

### Step 2: Simplify the expression
Using the approximations, the numerator becomes:

\[
2x^2 \cdot \frac{1}{x} - x \cdot \frac{1}{x} + \frac{1}{x} = 2x - 1 + \frac{1}{x}.
\]

The denominator becomes:

\[
x \cdot 1 = x.
\]

So, the expression simplifies to:

\[
\frac{2x - 1 + \frac{1}{x}}{x}.
\]

### Step 3: Evaluate the limit as \( x \to \infty \)
As \( x \to \infty \), the terms \( \frac{1}{x} \) approach zero. Therefore, the expression becomes:

\[
\frac{2x - 1}{x} = 2 - \frac{1}{x}.
\]

Again, as \( x \to \infty \), \( \frac{1}{x} \to 0 \), so the final result is:

\[
2.
\]

### Final Answer:
The value of the limit is \( \boxed{2} \).

---

Do you want any further details or have any questions?

Here are five related questions to deepen your understanding:
1. What is the limit of \( \frac{\sin(1/x)}{x} \) as \( x \to \infty \)?
2. How would the result change if the numerator had \( \tan\left(\frac{2}{x}\right) \) instead of \( \tan\left(\frac{1}{x}\right) \)?
3. How does the small-angle approximation work for higher powers of \( \frac{1}{x} \)?
4. What would the limit be if \( x \to 0 \) instead of \( x \to \infty \) in the given expression?
5. How do you apply L'Hopital's rule to limits involving trigonometric functions?

**Tip**: When solving limits involving trigonometric functions, small-angle approximations for \( \sin z \), \( \cos z \), and \( \tan z \) are very helpful when \( z \) approaches 0.

Learn how to evaluate the limit of the expression lim x -> ∞ (2x^2 * tan(1/x) - x * sin(1/x) + 1/x) / (x * cos(2/x)). This solution uses key mathematical concepts such as limits, small-angle approximations, and asymptotic behavior, making it suitable for advanced high school and early college students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation