Limit of x approaching infinity square root (x^4+25)/1-x^2

We are tasked with finding the limit of the following expression as \( x \) approaches infinity:

\[
\lim_{x \to \infty} \frac{\sqrt{x^4 + 25}}{1 - x^2}
\]

### Step-by-Step Solution:

1. **Simplifying the expression:**
   First, observe that as \( x \to \infty \), the highest power of \( x \) dominates both in the numerator and denominator. In this case, the numerator contains \( \sqrt{x^4 + 25} \), and the denominator contains \( 1 - x^2 \).

2. **Approximate the dominant term:**
   For large values of \( x \), \( x^4 + 25 \) is approximately \( x^4 \). Thus, we can approximate:
   \[
   \sqrt{x^4 + 25} \approx \sqrt{x^4} = x^2.
   \]

3. **Substitute into the limit:**
   So, the expression for large \( x \) becomes approximately:
   \[
   \frac{\sqrt{x^4 + 25}}{1 - x^2} \approx \frac{x^2}{1 - x^2}.
   \]

4. **Rewrite the expression:**
   Simplify the fraction:
   \[
   \frac{x^2}{1 - x^2} = \frac{x^2}{-x^2 + 1} = \frac{x^2}{-x^2 (1 - \frac{1}{x^2})}.
   \]
   As \( x \to \infty \), \( \frac{1}{x^2} \to 0 \), so this simplifies to:
   \[
   \frac{x^2}{-x^2} = -1.
   \]

5. **Conclusion:**
   Therefore, the limit of the given expression as \( x \to \infty \) is:
   \[
   \boxed{-1}.
   \]

Would you like more details or have any questions?

### Related questions:
1. How do we handle limits involving square roots of polynomials?
2. What is the behavior of functions dominated by different powers of \( x \) at infinity?
3. How can we find limits where both the numerator and denominator approach infinity?
4. Can we use L'Hopital's Rule in this situation to verify the result?
5. What are some other examples of indeterminate forms involving infinity?

**Tip:** When dealing with limits at infinity, always identify the dominant terms in both the numerator and denominator to simplify the expression.

Limit of x approaching infinity square root (x^4 + 25) / (1 - x^2)

Explore how to solve the limit problem involving sqrt(x^4 + 25) / (1 - x^2) as x approaches infinity. This solution includes algebraic simplification, handling dominant terms, and the behavior of limits at infinity. Suitable for Grades 11-12 and college students.

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