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The question is asking to evaluate the following limit as \( x \to \infty \):

\[
\lim_{x \to \infty} \sqrt{16x^2 + 10x - 3 - 4x + 1}
\]

To solve this, let's follow the steps:

### Step 1: Simplify the expression inside the square root.
We combine like terms inside the square root:

\[
16x^2 + 10x - 4x - 3 + 1 = 16x^2 + 6x - 2
\]

So the limit now becomes:

\[
\lim_{x \to \infty} \sqrt{16x^2 + 6x - 2}
\]

### Step 2: Factor out \( x^2 \) from inside the square root.
Since \( x \to \infty \), we factor \( x^2 \) from the terms inside the square root:

\[
\sqrt{16x^2 + 6x - 2} = \sqrt{x^2 \left(16 + \frac{6}{x} - \frac{2}{x^2}\right)}
\]

This simplifies to:

\[
\sqrt{x^2} \cdot \sqrt{16 + \frac{6}{x} - \frac{2}{x^2}} = |x| \cdot \sqrt{16 + \frac{6}{x} - \frac{2}{x^2}}
\]

Since \( x \to \infty \), we take \( |x| = x \) (because \( x \) is positive as it approaches infinity).

### Step 3: Evaluate the limit as \( x \to \infty \).
As \( x \to \infty \), the terms \( \frac{6}{x} \) and \( \frac{2}{x^2} \) approach 0. Therefore, the expression inside the square root approaches 16:

\[
\lim_{x \to \infty} x \cdot \sqrt{16 + \frac{6}{x} - \frac{2}{x^2}} = x \cdot \sqrt{16} = x \cdot 4 = 4x
\]

Thus, the limit is:

\[
\lim_{x \to \infty} 4x = 4
\]

### Step 4: Conclusion
It seems there might be a slight misinterpretation here, as none of the given options match exactly with this result. However, checking again, you may refer to the list of options carefully or look for details around the given choices.

Do you want further clarification on any step? Let me know!

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Here are 5 related questions:

1. How would the limit change if the leading coefficient of \( x^2 \) inside the square root were different?
2. What is the significance of factoring out \( x^2 \) in the limit calculation?
3. How can we handle limits where the highest power of \( x \) is not \( x^2 \)?
4. What happens if the sign of \( x \) changes in the expression?
5. How would you approach limits involving more complex roots or fractions?

Tip: Always simplify the highest-degree terms in the limit to make the calculation easier.

Evaluate the limit: lim_{x -> ∞} (sqrt(16x^2 + 10x - 3 - 4x + 1))

This problem explores the evaluation of a limit as x approaches infinity for a square root function involving a quadratic expression. Learn how to simplify the expression and apply the limit theorem to solve the problem.

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