To solve the limit:

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

Let's go step by step:

### Step 1: Substituting \( x = 1 \) directly
When we substitute \( x = 1 \), the expression becomes:

\[
(1 - 1) \sin(2\sqrt{1}) = 0 \times \sin(2) = 0
\]

This gives an indeterminate form \( 0 \times \sin(2) = 0 \). We need a more careful evaluation, such as using L'Hopital's Rule or another method.

### Step 2: Using L'Hopital's Rule
L'Hopital's Rule applies to indeterminate forms like \( 0 \times \infty \) or \( \frac{0}{0} \). We can rewrite the expression to apply it:

\[
\lim_{{x \to 1}} (x - 1) \sin(2\sqrt{x}) = \lim_{{x \to 1}} \frac{\sin(2\sqrt{x})}{\frac{1}{x - 1}}
\]

We now have the indeterminate form \( \frac{0}{0} \), so we can apply L'Hopital's Rule by differentiating the numerator and denominator:

#### Derivative of the numerator:
\[
\frac{d}{dx} \left( \sin(2\sqrt{x}) \right) = \cos(2\sqrt{x}) \cdot \frac{d}{dx} (2\sqrt{x}) = \cos(2\sqrt{x}) \cdot \frac{1}{\sqrt{x}}
\]

#### Derivative of the denominator:
\[
\frac{d}{dx} \left( \frac{1}{x - 1} \right) = -\frac{1}{(x - 1)^2}
\]

### Step 3: Applying L'Hopital's Rule
Now, evaluate the limit:

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

This limit should simplify to 0. Therefore, the original limit is:

\[
\lim_{{x \to 1}} (x - 1) \sin(2\sqrt{x}) = 0
\]

Let me know if you'd like more details, or if you have any further questions!

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Here are 5 related questions for further practice:
1. What is the limit \( \lim_{{x \to 0}} x \sin\left(\frac{1}{x}\right) \)?
2. How does L'Hopital's Rule apply to indeterminate forms?
3. What is the derivative of \( \sin(3x^2) \)?
4. Can you find the limit of \( \lim_{{x \to 1}} \frac{x^2 - 1}{x - 1} \)?
5. What is the Taylor expansion of \( \sin(x) \) around \( x = 0 \)?

**Tip:** Always check if the limit gives an indeterminate form like \( \frac{0}{0} \) or \( \infty \times 0 \) before applying L'Hopital's Rule.

Learn how to solve the limit lim x→1 (x − 1) sin(2√x) using L'Hopital's Rule. This solution includes key concepts such as trigonometric limits, derivatives, and applying L'Hopital's Rule for indeterminate forms. Suitable for students in undergraduate calculus.

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