Math Problem Statement

We can now find the limit of ln(y) by multiplying the numerator and denominator by x^2.

Solution

To solve this limit problem, let’s analyze each part step-by-step:

  1. Expression Overview: We are given: limx(11+5x5x25x+115x2)x2x2\lim_{x \to \infty} \left( \frac{\frac{1}{1 + \frac{5}{x}} - \frac{5}{x^2}}{\frac{5}{x} + 1 - \frac{1}{5x^2}} \right) \cdot \frac{x^2}{x^2} The goal is to simplify the limit as xx \to \infty by multiplying both the numerator and the denominator by x2x^2.

  2. Multiplying by x2x^2:

    • When multiplying the numerator by x2x^2: x2(11+5x5x2)=x21+5x5x^2 \left(\frac{1}{1 + \frac{5}{x}} - \frac{5}{x^2}\right) = \frac{x^2}{1 + \frac{5}{x}} - 5
    • When multiplying the denominator by x2x^2: x2(5x+115x2)=5x+x215x^2 \left(\frac{5}{x} + 1 - \frac{1}{5x^2}\right) = 5x + x^2 - \frac{1}{5}
  3. Evaluating Each Term as xx \to \infty:

    • In the numerator: x21+5x5x25\frac{x^2}{1 + \frac{5}{x}} - 5 \approx x^2 - 5 because 5x0\frac{5}{x} \to 0 as xx \to \infty.
    • In the denominator: 5x+x215x25x + x^2 - \frac{1}{5} \approx x^2 since 5x/x205x/x^2 \to 0 and 15x20\frac{1}{5x^2} \to 0.
  4. Taking the Limit: As xx \to \infty, the expression simplifies to: limxx25x2=limx(15x2)\lim_{x \to \infty} \frac{x^2 - 5}{x^2} = \lim_{x \to \infty} \left(1 - \frac{5}{x^2}\right) Since 5x20\frac{5}{x^2} \to 0 as xx \to \infty, the limit is: 11

So, the answer for the limit is 1.

Would you like more details on any specific part of this solution?


Related Questions

  1. What is the significance of multiplying by x2x^2 in limit problems?
  2. How would this approach change if we were dealing with a limit as x0x \to 0 instead?
  3. What other techniques are commonly used to simplify limits at infinity?
  4. Can you apply L'Hôpital's Rule to this type of problem? Why or why not?
  5. How does the limit change if additional higher-degree terms are in the numerator?

Tip

When simplifying limits at infinity, always consider dividing by the highest power of xx in the expression to reveal the dominant terms as xx \to \infty.

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Math Problem Analysis

Mathematical Concepts

Limits
Infinity
Rational Expressions

Formulas

Limit as x approaches infinity
Multiplication by x^2 for simplification

Theorems

Limit Laws at Infinity

Suitable Grade Level

Grades 11-12