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We are tasked with finding the following limit:

\[
\lim_{h \to 0^+} \frac{\sqrt{h^2 + 4h + 3} - \sqrt{3}}{h}
\]

### Step-by-Step Solution:

#### Step 1: Recognize the Indeterminate Form
As \( h \to 0^+ \), directly substituting \( h = 0 \) in the expression gives:

\[
\frac{\sqrt{0^2 + 4(0) + 3} - \sqrt{3}}{0} = \frac{\sqrt{3} - \sqrt{3}}{0} = \frac{0}{0}
\]

This is an indeterminate form, which suggests that we need to manipulate the expression further.

#### Step 2: Multiply by the Conjugate
To resolve the indeterminate form, multiply both the numerator and denominator by the conjugate of the numerator. The conjugate of \( \sqrt{h^2 + 4h + 3} - \sqrt{3} \) is \( \sqrt{h^2 + 4h + 3} + \sqrt{3} \).

Thus, multiply the expression by:

\[
\frac{\sqrt{h^2 + 4h + 3} + \sqrt{3}}{\sqrt{h^2 + 4h + 3} + \sqrt{3}}
\]

This gives:

\[
\lim_{h \to 0^+} \frac{\left( \sqrt{h^2 + 4h + 3} - \sqrt{3} \right)\left( \sqrt{h^2 + 4h + 3} + \sqrt{3} \right)}{h \left( \sqrt{h^2 + 4h + 3} + \sqrt{3} \right)}
\]

#### Step 3: Simplify the Numerator
The numerator simplifies as follows, using the difference of squares formula \( (a - b)(a + b) = a^2 - b^2 \):

\[
\left( \sqrt{h^2 + 4h + 3} \right)^2 - \left( \sqrt{3} \right)^2 = (h^2 + 4h + 3) - 3 = h^2 + 4h
\]

So the expression becomes:

\[
\lim_{h \to 0^+} \frac{h^2 + 4h}{h \left( \sqrt{h^2 + 4h + 3} + \sqrt{3} \right)}
\]

#### Step 4: Factor and Simplify
Factor the numerator:

\[
h^2 + 4h = h(h + 4)
\]

Now the expression simplifies to:

\[
\lim_{h \to 0^+} \frac{h(h + 4)}{h \left( \sqrt{h^2 + 4h + 3} + \sqrt{3} \right)}
\]

Cancel the \( h \) terms:

\[
\lim_{h \to 0^+} \frac{h + 4}{\sqrt{h^2 + 4h + 3} + \sqrt{3}}
\]

#### Step 5: Evaluate the Limit
As \( h \to 0^+ \), substitute \( h = 0 \) into the remaining expression:

\[
\frac{0 + 4}{\sqrt{0^2 + 4(0) + 3} + \sqrt{3}} = \frac{4}{\sqrt{3} + \sqrt{3}} = \frac{4}{2\sqrt{3}} = \frac{2}{\sqrt{3}}
\]

To rationalize the denominator, multiply by \( \frac{\sqrt{3}}{\sqrt{3}} \):

\[
\frac{2}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{2\sqrt{3}}{3}
\]

Thus, the final value of the limit is:

\[
\boxed{\frac{2\sqrt{3}}{3}}
\]

Let me know if you'd like further details or clarifications!

### Follow-up Questions:
1. How does multiplying by the conjugate help in resolving indeterminate forms?
2. Why does the indeterminate form \( \frac{0}{0} \) occur frequently in limits involving square roots?
3. What other techniques, besides multiplying by the conjugate, can be used to resolve limits with square roots?
4. How would the solution differ if the limit were approached as \( h \to 0^- \)?
5. Can you explain the importance of rationalizing the denominator in the final step?

### Tip:
When faced with limits involving square roots, multiplying by the conjugate is often a powerful tool to simplify the expression and resolve indeterminate forms.

Find the limit of the expression as h approaches 0 from the positive direction: (sqrt(h^2 + 4h + 3) - sqrt(3)) / h

Discover the step-by-step process of finding the limit of (sqrt(h^2 + 4h + 3) - sqrt(3)) / h as h approaches 0+ using the conjugate method. This calculus solution uses the difference of squares identity to resolve the indeterminate form and rationalizes the denominator.

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