We are tasked with calculating the limit of the function $f(x) = \sqrt{144x^2 + x - 12x}$ as $x \to \infty$.

The given function simplifies to: $f(x) = \sqrt{144x^2 + x - 12x} = \sqrt{144x^2 - 11x}.$

Step-by-step approach to solving the limit:

1. The expression inside the square root is dominated by the $144x^2$ term as $x \to \infty$, so the behavior of the function for large $x$ will mainly depend on $\sqrt{144x^2}$.

2. Factor out $x^2$ from the expression inside the square root to simplify the limit: $f(x) = \sqrt{x^2 \left( 144 - \frac{11}{x} \right)}.$

3. Simplify further: $f(x) = x \sqrt{144 - \frac{11}{x}}.$

4. As $x \to \infty$, the term $\frac{11}{x} \to 0$, so the expression becomes: $f(x) \to x \cdot \sqrt{144} = x \cdot 12.$

5. Therefore, as $x \to \infty$, the limit is: $\lim_{x \to \infty} f(x) = 12x \to \infty.$

Thus, the limit of the function $f(x)$ as $x \to \infty$ is infinite.

Would you like more details on any step, or do you have any questions about the process?

Related Questions:

1. How would the result change if the leading term inside the square root were different?
2. What happens to the limit if the $-11x$ term is positive instead of negative?
3. How does factoring out the dominant term help simplify limits like these?
4. Can you apply L'Hopital's Rule to verify this result?
5. What other techniques can be used to evaluate limits of functions involving square roots?

Tip:

When evaluating limits of functions with square roots, factor out the highest power of $x$ to simplify the expression.