limℎ→02+ℎ⎯⎯⎯⎯⎯⎯⎯⎯⎯√−11ℎ

The given expression is a limit problem:

\[
\lim_{h \to 0} \frac{\sqrt{2 + h} - 1}{h}
\]

To evaluate this limit, we can apply a technique known as **rationalizing the numerator**. Here's a step-by-step explanation:

### Step 1: Multiply by the conjugate
The conjugate of \(\sqrt{2 + h} - 1\) is \(\sqrt{2 + h} + 1\). Multiply both the numerator and the denominator by this conjugate:

\[
\frac{\sqrt{2 + h} - 1}{h} \times \frac{\sqrt{2 + h} + 1}{\sqrt{2 + h} + 1} = \frac{\left(\sqrt{2 + h} - 1\right)\left(\sqrt{2 + h} + 1\right)}{h \left(\sqrt{2 + h} + 1\right)}
\]

### Step 2: Simplify the numerator
By using the difference of squares formula \((a - b)(a + b) = a^2 - b^2\), the numerator becomes:

\[
\left(\sqrt{2 + h}\right)^2 - 1^2 = (2 + h) - 1 = h
\]

So the expression simplifies to:

\[
\frac{h}{h \left(\sqrt{2 + h} + 1\right)} = \frac{1}{\sqrt{2 + h} + 1}
\]

### Step 3: Take the limit as \( h \to 0 \)
Now, substitute \( h = 0 \) into the simplified expression:

\[
\frac{1}{\sqrt{2 + 0} + 1} = \frac{1}{\sqrt{2} + 1}
\]

Thus, the limit is:

\[
\boxed{\frac{1}{\sqrt{2} + 1}}
\]

### Conclusion
The value of the limit is \(\frac{1}{\sqrt{2} + 1}\).

Would you like more details on any specific step, or do you have any questions?

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Here are 5 related questions to explore:

1. What is the conjugate of a binomial expression, and how is it used in calculus?
2. How do you handle limits that involve square roots or other irrational expressions?
3. What is the difference between rationalizing the numerator and rationalizing the denominator?
4. How does the difference of squares formula work in simplifying expressions?
5. How can L'Hopital's Rule be applied to indeterminate forms like \(\frac{0}{0}\)?

**Tip:** When evaluating limits involving square roots, rationalizing the numerator can often simplify the expression significantly.

Solve the limit problem lim h→0 (√(2+h) - 1)/h using rationalization of the numerator. This detailed step-by-step solution covers key calculus concepts, including limits and the difference of squares formula. Suitable for high school students in Grades 11-12.

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