We are tasked with calculating the limit:

$\lim_{x \to \infty} f(x) \quad \text{where} \quad f(x) = \sqrt{144x^2 + x} - 12x$

Step 1: Factor out $x^2$ from inside the square root.

We begin by factoring $x^2$ from the expression inside the square root:

$f(x) = \sqrt{x^2(144 + \frac{1}{x})} - 12x$

Simplifying the square root:

$f(x) = x\sqrt{144 + \frac{1}{x}} - 12x$

Step 2: Expand the square root using a binomial approximation for large $x$.

As $x \to \infty$, $\frac{1}{x}$ becomes very small, so we can use the binomial expansion for $\sqrt{144 + \frac{1}{x}}$ around $\frac{1}{x} = 0$. The binomial expansion of $\sqrt{1 + u}$ is approximately $1 + \frac{u}{2}$ for small $u$.

Here, $u = \frac{1}{144x}$, so:

$\sqrt{144 + \frac{1}{x}} \approx 12 + \frac{1}{24x}$

Step 3: Substitute the approximation into the original expression.

Substitute $\sqrt{144 + \frac{1}{x}}$ back into the expression for $f(x)$:

$f(x) \approx x\left(12 + \frac{1}{24x}\right) - 12x$

Step 4: Simplify the expression.

Now simplify:

$f(x) \approx 12x + \frac{x}{24x} - 12x$

The $12x$ terms cancel out, leaving:

$f(x) \approx \frac{1}{24}$

Step 5: Take the limit as $x \to \infty$.

Since the remaining expression is constant as $x \to \infty$, we find:

$\lim_{x \to \infty} f(x) = \frac{1}{24}$

Thus, the exact value of the limit is:

$\boxed{\frac{1}{24}}$

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Would you like more details or have any questions? Here are some related questions:

1. How does the binomial approximation work for other types of functions?
2. What if we expand further terms in the binomial expansion—would the limit change?
3. How can we rigorously justify ignoring higher-order terms in limits as $x \to \infty$?
4. What are other techniques for solving limits involving square roots?
5. Can you find the limit if the function inside the square root had a different power of $x$?

Tip: When solving limits at infinity, look for dominant terms that grow the fastest as $x \to \infty$, and simplify expressions accordingly.