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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.

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From the graph of \(f\), evaluate

\[\Large \lim_{x\to6}f(f(x)).\]

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Micah Wood**

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Some people claim that \( 0 ^ 0 = 1 \). What is

\[ \Large \lim_{ x \rightarrow 0^+ } x ^ { \frac{ 1}{\ln x} }?\]

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Calvin Lin**

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A man stuck in a small sailboat on a perfectly calm lake throws a stone overboard. It sinks to the bottom of the lake.

When the water again settles to a perfect calm, is the water level in the lake higher, lower, or in the same place compared to where it was before the stone was cast in?

*Hint: You can use limits to solve this problem!*

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Zandra Vinegar**

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Evaluate

\[ \lim_{x \rightarrow 0^+ } \sqrt{ x + \sqrt{ x + \sqrt{ x + \ldots } } }. \]

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Calvin Lin**

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\[\large \lim_{x \to 1} \left( \frac{23}{1-x^{23}}-\frac{11}{1-x^{11}} \right) = \ ?\]

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Kazem Sepehrinia**

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