https://r2.erweima.ai/mathbot/image/172676802634631b63vqq.jpg

The expression provided in the image appears to be a limit problem. Here's the mathematical expression extracted:

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

### Step-by-step solution:

1. **Analyze the denominator:**
   \[
   x^2 - 4x + 4 = (x - 2)^2
   \]
   The denominator simplifies to \((x - 2)^2\).

2. **Substitute the simplified denominator:**
   \[
   \frac{(2x - 4)\sqrt{x^2 + 1}}{(x - 2)^2}
   \]
   Notice that \(2x - 4 = 2(x - 2)\).

3. **Rewrite the expression:**
   \[
   \frac{2(x - 2)\sqrt{x^2 + 1}}{(x - 2)^2}
   \]
   Cancel one \((x - 2)\) term from both the numerator and denominator:
   \[
   \frac{2\sqrt{x^2 + 1}}{x - 2}
   \]

4. **Substitute \(x = 2\):**
   Direct substitution would result in a zero denominator, so we need to check if this is an indeterminate form or if there's another method, such as L'Hopital's Rule or expansion.

   - Applying **L'Hopital's Rule** (since both the numerator and denominator tend to zero as \(x \to 2\)):

   Differentiate the numerator:
   \[
   \frac{d}{dx}[2\sqrt{x^2 + 1}] = \frac{2x}{\sqrt{x^2 + 1}}
   \]

   Differentiate the denominator:
   \[
   \frac{d}{dx}[x - 2] = 1
   \]

   So, the new limit is:
   \[
   \lim_{x \to 2} \frac{2x}{\sqrt{x^2 + 1}} = \frac{2(2)}{\sqrt{2^2 + 1}} = \frac{4}{\sqrt{5}}
   \]

   Thus, the final answer is:
   \[
   \frac{4}{\sqrt{5}}
   \]

Would you like more details or have any questions?

### Related Questions:
1. How does L'Hopital's Rule work in cases where both the numerator and denominator approach zero?
2. Can we solve this limit without using L'Hopital's Rule, perhaps by factoring or simplifying differently?
3. Why do we cancel out terms like \((x - 2)\) in limit problems?
4. How would this limit change if the square root term had a different expression inside it?
5. What happens if we apply a series expansion to the square root term?

### Tip:
Always check whether you are dealing with an indeterminate form before applying L'Hopital's Rule, as it only applies in those cases.

Evaluate the limit: lim(x -> 2) [(2x - 4)√(x^2 + 1)] / (x^2 - 4x + 4)

Explore the solution to the limit problem lim(x -> 2) [(2x - 4)√(x^2 + 1)] / (x^2 - 4x + 4) using L'Hopital's Rule. This problem highlights key calculus concepts like limit evaluation, algebraic simplification, and the application of differentiation for indeterminate forms.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation