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The problem asks to evaluate the limits of the function \( f(x) = 2x + \sqrt{6x^2 - 2} \) as \( x \) approaches positive infinity (\(\infty\)) and negative infinity (\(-\infty\)).

Let's break it down step by step.

### 1. Limit as \( x \) approaches \( +\infty \):
\[
\lim_{x \to +\infty} f(x) = \lim_{x \to +\infty} \left( 2x + \sqrt{6x^2 - 2} \right)
\]

First, notice that as \( x \) becomes very large, \( 6x^2 - 2 \) behaves very similarly to \( 6x^2 \). So, the square root \( \sqrt{6x^2 - 2} \) behaves like \( \sqrt{6x^2} = \sqrt{6} \cdot |x| \).

Since \( x \) is approaching \( +\infty \), \( |x| = x \), so:
\[
\lim_{x \to +\infty} \sqrt{6x^2 - 2} = \lim_{x \to +\infty} \sqrt{6} \cdot x = \sqrt{6} \cdot x
\]

Thus:
\[
\lim_{x \to +\infty} f(x) = \lim_{x \to +\infty} \left( 2x + \sqrt{6} \cdot x \right) = \lim_{x \to +\infty} \left( (2 + \sqrt{6})x \right) = +\infty
\]

### 2. Limit as \( x \) approaches \( -\infty \):
\[
\lim_{x \to -\infty} f(x) = \lim_{x \to -\infty} \left( 2x + \sqrt{6x^2 - 2} \right)
\]

Again, using the approximation \( \sqrt{6x^2 - 2} \approx \sqrt{6} \cdot |x| \), but this time as \( x \) approaches \( -\infty \), \( |x| = -x \).

So:
\[
\lim_{x \to -\infty} \sqrt{6x^2 - 2} = \lim_{x \to -\infty} \sqrt{6} \cdot (-x) = -\sqrt{6} \cdot x
\]

Thus:
\[
\lim_{x \to -\infty} f(x) = \lim_{x \to -\infty} \left( 2x - \sqrt{6} \cdot x \right) = \lim_{x \to -\infty} \left( (2 - \sqrt{6})x \right)
\]

Here, \( 2 - \sqrt{6} \) is a negative number (approximately -0.45), so:
\[
\lim_{x \to -\infty} f(x) = -\infty
\]

### Summary of Answers:
- \(\lim_{x \to +\infty} f(x) = +\infty\)
- \(\lim_{x \to -\infty} f(x) = -\infty\)

Would you like further details on the steps, or do you have any additional questions?

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Here are five related questions you might consider exploring:

1. How would the function \( f(x) = 2x + \sqrt{6x^2 - 2} \) behave if the square root term was instead \( \sqrt{6x^2 + 2} \)?
2. What is the significance of the coefficient \( \sqrt{6} \) in this context?
3. How would you find the horizontal asymptotes of \( f(x) \) for other similar functions?
4. Can you graph \( f(x) \) to visually confirm the limits?
5. What happens to the function \( f(x) = 2x + \sqrt{6x^2 - 2} \) as \( x \) approaches zero?

**Tip:** When dealing with limits involving square roots, factor out the highest power of \( x \) to simplify the expression.

For the function f(x) given below, evaluate lim_{x→∞} f(x) and lim_{x→-∞} f(x).

f(x) = 2x + √(6x² - 2)

Provide your answer below:

lim_{x→∞} f(x) = ___

lim_{x→-∞} f(x) = ___

Learn how to evaluate the limits of the function f(x) = 2x + √(6x² - 2) as x approaches both positive and negative infinity. This problem involves key mathematical concepts such as limits, asymptotic behavior, and square roots, suitable for high school students in Grades 10-12.

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