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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. See more

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\[ \boxed{\begin{array}{c|r:r:r:r:r} x & 0.1 & 0.01& 0.001& 0.0001& 0.00001\\ \hline \lfloor x \rfloor & 0 & 0 & 0 & 0 & 0 \end{array}} \]

By looking at the table above, is it true that as \(x\) approaches 0, then \(\lfloor x \rfloor \) approaches 0 as well?
That is, is \( \displaystyle \lim_{x\to0} \lfloor x\rfloor = 0 \) correct?

\[\] Notation: \( \lfloor \cdot \rfloor \) denotes the floor function.

\[ \Large \lim_{x\to0} \dfrac{x}{x!} = \, ? \]

Note: Treat \( x! = \Gamma(x+1) \).

\[ \lim_{x\to\infty} \dfrac{\sin x}x = \, ? \]

Find \( \displaystyle\lim_{x\rightarrow 3} \) \( \dfrac{x^2-9}{x-3} \)

\[\large \lim_{x \to 1} \dfrac{|x-1|}{x-1} = \, ? \]

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