We need to evaluate the following limits:

$\lim_{x \to \infty} \frac{4x+2}{|x|+6x}$

$\lim_{x \to -\infty} \frac{4x+2}{|x|+6x}$

Step 1: Consider $x \to \infty$

For $x \to \infty$, we know that $|x| = x$, so the given function simplifies to:

$\frac{4x+2}{x+6x} = \frac{4x+2}{7x}$

Now, divide both numerator and denominator by $x$:

$\frac{4 + \frac{2}{x}}{7}$

As $x \to \infty$, the term $\frac{2}{x} \to 0$, so we get:

$\frac{4}{7}$

Step 2: Consider $x \to -\infty$

For $x \to -\infty$, we know that $|x| = -x$, so the function becomes:

$\frac{4x+2}{-x+6x} = \frac{4x+2}{5x}$

Dividing both numerator and denominator by $x$:

$\frac{4 + \frac{2}{x}}{5}$

As $x \to -\infty$, the term $\frac{2}{x} \to 0$, so we get:

$\frac{4}{5}$

Final Answers:

$\lim_{x \to \infty} \frac{4x+2}{|x|+6x} = \frac{4}{7}$

$\lim_{x \to -\infty} \frac{4x+2}{|x|+6x} = \frac{4}{5}$

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Would you like a detailed explanation of any step?

Related Questions:

1. What happens if $|x|$ is replaced by a different function, such as $x^2$?
2. How do we handle limits involving absolute values in more complex expressions?
3. What if we had a function of the form $(ax+b)/(|x|+cx)$? How does the limit depend on $a$ and $c$?
4. How would the result change if we considered one-sided limits instead?
5. How does the presence of higher-degree terms in the numerator or denominator affect the limit?

Tip:

When solving limits involving absolute values, always consider separate cases for $x > 0$ and $x < 0$ to avoid incorrect simplifications.