Limits of Functions

Limits of Functions: Level 3 Challenges


limnx+2x+3x++nxn2\large \displaystyle\lim_{n \to \infty} \dfrac{\lfloor x \rfloor+\lfloor 2x \rfloor+\lfloor 3x \rfloor+\cdots+\lfloor nx \rfloor}{n^2}

Let xx be a constant real number. Find the value of the limit above in terms of xx.

Notation: \lfloor \cdot \rfloor denotes the floor function.

From the graph of ff, evaluate

limx6f(f(x)).\Large \lim_{x\to6}f(f(x)).

Some people claim that 00=1 0 ^ 0 = 1 . What is

limx0+x1lnx? \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!


limx0+x+x+x+. \lim_{x \rightarrow 0^+ } \sqrt{ x + \sqrt{ x + \sqrt{ x + \ldots } } }.

limx1(231x23111x11)=?\large \lim_{x \to 1} \left( \frac{23}{1-x^{23}}-\frac{11}{1-x^{11}} \right) = \, ?


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