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# Limits of Functions

What happens when a function's output isn't calculable directly – e.g., at infinity – but we still need to understand its behavior? That's where limits come in.

If \[ \lim_{x \to 4} \frac{f(x) - 5}{x-2}=1, \] find \( \displaystyle \lim_{x \to 4} f(x). \)

Find the value of

\[ \lim_{x \to \infty} \left( \sqrt{x^2 + \lfloor x \rfloor} - x \right). \]

**Assumptions and Details:**

\( \lfloor \cdot \rfloor \) is the floor function such that \( \lfloor x \rfloor = n \) if \( n \leq x < n+1, \) where \( n \) is an integer.

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