Math Problem Statement

lim x→∞ x(√(x² + 2x) - 2√(x² + x))

Solution

The given limit problem is:

limxx(x2+2x2x2+x)\lim_{x \to \infty} x \left( \sqrt{x^2 + 2x} - 2 \sqrt{x^2 + x} \right)

To solve this, let's simplify the expression inside the limit step by step.

Step 1: Approximate the square roots for large xx

When xx \to \infty, the dominant term inside the square roots is x2x^2. Thus, we can factor x2x^2 from the terms inside the square roots:

x2+2x=x1+2xandx2+x=x1+1x\sqrt{x^2 + 2x} = x \sqrt{1 + \frac{2}{x}} \quad \text{and} \quad \sqrt{x^2 + x} = x \sqrt{1 + \frac{1}{x}}

For large xx, we can use the first two terms of the binomial expansion for square roots:

1+2x1+1xand1+1x1+12x\sqrt{1 + \frac{2}{x}} \approx 1 + \frac{1}{x} \quad \text{and} \quad \sqrt{1 + \frac{1}{x}} \approx 1 + \frac{1}{2x}

Thus:

x2+2xx(1+1x)=x+1\sqrt{x^2 + 2x} \approx x \left( 1 + \frac{1}{x} \right) = x + 1 x2+xx(1+12x)=x+12\sqrt{x^2 + x} \approx x \left( 1 + \frac{1}{2x} \right) = x + \frac{1}{2}

Step 2: Substitute into the expression

Now substitute these approximations into the limit expression:

x((x+1)2(x+12))x \left( (x + 1) - 2(x + \frac{1}{2}) \right)

Simplify the expression inside the parentheses:

(x+1)2(x+12)=x+12x1=x(x + 1) - 2(x + \frac{1}{2}) = x + 1 - 2x - 1 = -x

Thus, the expression becomes:

x(x)=x2x \cdot (-x) = -x^2

Step 3: Evaluate the limit

The limit now becomes:

limxx2\lim_{x \to \infty} -x^2

As xx \to \infty, this tends to -\infty.

Final Answer:

\boxed{-\infty}

Would you like more details on any specific step? Here are 5 related questions to further explore this concept:

  1. How does binomial expansion help in simplifying square roots?
  2. What happens if we ignore the second terms in the approximations for the square roots?
  3. How can this method be generalized for similar limits involving square roots?
  4. What is the role of factoring the dominant term when simplifying limits?
  5. Could L'Hopital's Rule be used as an alternative approach here?

Tip: When simplifying square roots for large xx, factor out the highest power of xx inside the square root and approximate using binomial expansion.

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Math Problem Analysis

Mathematical Concepts

Limits
Binomial Expansion
Asymptotic Approximation
Algebra

Formulas

√(x² + ax) ≈ x(1 + a/(2x)) for large x
lim x→∞ f(x)

Theorems

Binomial Expansion
Limit Rules for Infinity

Suitable Grade Level

College-level calculus