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The given limit problem is:

\[
\lim_{{x \to 1^+}} \left( \frac{1}{\sqrt{x-1}} - \frac{1}{\sqrt{x^2-1}} \right)
\]

Let's analyze the limit step by step.

1. **Simplification of the second term:**
   The expression inside the second square root is \(x^2 - 1\), which can be factored:
   \[
   x^2 - 1 = (x - 1)(x + 1)
   \]
   Thus, the second term becomes:
   \[
   \frac{1}{\sqrt{x^2 - 1}} = \frac{1}{\sqrt{(x-1)(x+1)}} = \frac{1}{\sqrt{x-1} \cdot \sqrt{x+1}}
   \]

2. **Rewrite the original expression:**
   Substituting this into the original limit, we have:
   \[
   \lim_{{x \to 1^+}} \left( \frac{1}{\sqrt{x-1}} - \frac{1}{\sqrt{x-1} \cdot \sqrt{x+1}} \right)
   \]

3. **Factor out \(\frac{1}{\sqrt{x-1}}\):**
   \[
   \lim_{{x \to 1^+}} \frac{1}{\sqrt{x-1}} \left( 1 - \frac{1}{\sqrt{x+1}} \right)
   \]

4. **Examine the limit as \(x \to 1^+\):**
   - As \(x \to 1^+\), \(\sqrt{x - 1} \to 0^+\).
   - \(\sqrt{x+1} \to \sqrt{2}\).

   So, the expression becomes:
   \[
   \lim_{{x \to 1^+}} \frac{1}{0^+} \left( 1 - \frac{1}{\sqrt{2}} \right)
   \]
   The term inside the parentheses approaches a constant value:
   \[
   1 - \frac{1}{\sqrt{2}} = 1 - \frac{\sqrt{2}}{2} \approx 1 - 0.7071 = 0.2929
   \]

5. **Conclusion:**
   The expression simplifies to:
   \[
   \frac{1}{0^+} \cdot 0.2929
   \]
   Since \(\frac{1}{0^+}\) tends to infinity, the overall limit tends to infinity. Therefore:

\[
\boxed{\infty}
\]

The limit as \(x\) approaches 1 from the right is infinity.

Would you like more details on any step or have further questions?

Here are some related questions:
1. How do you factor and simplify expressions like \(x^2 - 1\)?
2. What is the meaning of \(x \to 1^+\) in limit problems?
3. How does multiplying by \(\frac{1}{0^+}\) lead to infinity?
4. How do square roots affect limit behavior near boundary points?
5. What strategies can simplify expressions involving square roots?

**Tip:** In limits involving indeterminate forms or complex expressions, factoring and cancellation often help to simplify the problem!

Evaluate the limit: lim (x → 1⁺) (1 / √(x - 1) - 1 / √(x² - 1))

Learn how to solve the limit problem lim (x → 1⁺) (1 / √(x - 1) - 1 / √(x² - 1)) with a step-by-step explanation, including key algebraic techniques, the difference of squares formula, and the behavior of functions near boundary points. Suitable for high school students in Grades 11-12.

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