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déterminer la limite de la fonction suivante

We are tasked with finding the limit of the following function as \( x \to 2 \):

\[
f(x) = \frac{\sqrt{x^3 - 12x + 17} - \sqrt{x^3 - 3x^2 + 5}}{x^2 - 6x + 4}
\]
with \( x = 2 \).

### Step 1: Analyze the Denominator
First, we simplify the denominator:

\[
x^2 - 6x + 4
\]

Substituting \( x = 2 \):

\[
2^2 - 6(2) + 4 = 4 - 12 + 4 = -4
\]

So, the denominator is \( -4 \) when \( x = 2 \).

### Step 2: Analyze the Numerator
We evaluate each square root term in the numerator separately by substituting \( x = 2 \):

- First term: \( \sqrt{x^3 - 12x + 17} \)

\[
\sqrt{2^3 - 12(2) + 17} = \sqrt{8 - 24 + 17} = \sqrt{1} = 1
\]

- Second term: \( \sqrt{x^3 - 3x^2 + 5} \)

\[
\sqrt{2^3 - 3(2^2) + 5} = \sqrt{8 - 12 + 5} = \sqrt{1} = 1
\]

### Step 3: Compute the Numerator
Now we subtract the two terms:

\[
\sqrt{x^3 - 12x + 17} - \sqrt{x^3 - 3x^2 + 5} = 1 - 1 = 0
\]

### Step 4: Evaluate the Limit
Since the numerator is 0 and the denominator is \( -4 \), the value of the function at \( x = 2 \) is:

\[
f(2) = \frac{0}{-4} = 0
\]

Thus, the limit of the function as \( x \to 2 \) is:

\[
\boxed{0}
\]

Would you like any further clarification or details?

### Related Questions:
1. What methods can be used to evaluate limits with indeterminate forms?
2. How do you find the derivative of a function involving square roots?
3. What is the significance of the limit of a function in calculus?
4. Can you explain how to simplify complex rational expressions?
5. How does one evaluate a limit using L'Hôpital's Rule?

### Tip:
When dealing with square roots in limits, rationalizing the numerator or denominator can often help simplify the expression.

Determine the limit of the function sqrt(x^3 - 12x + 17) - sqrt(x^3 - 3x^2 + 5) / (x^2 - 6x + 4) at x = 2.

Learn how to find the limit of the function involving radicals and algebraic expressions, where sqrt(x^3 - 12x + 17) - sqrt(x^3 - 3x^2 + 5) is divided by (x^2 - 6x + 4) at x = 2. Detailed steps provided.

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