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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

# Level 3

From the graph of $$f$$, evaluate

$\Large \lim_{x\to6}f(f(x)).$

Some people claim that $$0 ^ 0 = 1$$. What is

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

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!

Evaluate

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

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

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