The limit as x approaches infinity of (x)/(square root (x^2+1))

We are asked to find the limit of the following expression as \( x \) approaches infinity:

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

### Step-by-Step Solution:

1. **Factor \( x^2 \) out of the square root:**

   Rewrite the expression inside the square root:
   
   \[
   \sqrt{x^2 + 1} = \sqrt{x^2(1 + \frac{1}{x^2})} = x\sqrt{1 + \frac{1}{x^2}}
   \]

   So the expression becomes:

   \[
   \frac{x}{\sqrt{x^2 + 1}} = \frac{x}{x \sqrt{1 + \frac{1}{x^2}}} = \frac{1}{\sqrt{1 + \frac{1}{x^2}}}
   \]

2. **Evaluate the limit as \( x \to \infty \):**

   As \( x \to \infty \), the term \( \frac{1}{x^2} \to 0 \). Therefore, the expression simplifies to:
   
   \[
   \lim_{{x \to \infty}} \frac{1}{\sqrt{1 + \frac{1}{x^2}}} = \frac{1}{\sqrt{1 + 0}} = 1
   \]

Thus, the limit is:

\[
\boxed{1}
\]

Would you like more details on this solution, or do you have any further questions?

Here are 5 related questions to explore:

1. What is the limit of \( \frac{x^2}{\sqrt{x^4 + 1}} \) as \( x \to \infty \)?
2. How do you handle limits involving radicals and infinity in general?
3. What is the limit of \( \frac{x}{\sqrt{x^2 + x}} \) as \( x \to \infty \)?
4. Can L'Hopital's Rule be applied to find the limit here? If so, how?
5. How does the behavior of functions change as \( x \to -\infty \)?

**Tip:** When finding limits involving square roots at infinity, factor out the highest power of \( x \) to simplify the expression.

Explore the detailed step-by-step solution to finding the limit of (x)/(sqrt(x^2 + 1)) as x approaches infinity. Learn key concepts involving radicals and limits at infinity, suitable for high school students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation