Calculus

Relative Magnitude of Functions

If $x$ goes to infinity, which of the following is the largest of all:

$\begin{array}{c}&\frac{x-3}{3x}, &3\sin\frac{1}{x}, &\cos \frac{1}{x}? \end{array}$

Which of the following is larger in the limit as $x$ grows to infinity?

$\begin{array}{c}&\sqrt{\lfloor 4x^2+9x \rfloor}, &2x \end{array}$

Details and assumptions

The function $\lfloor x \rfloor: \mathbb{R} \rightarrow \mathbb{Z}$ refers to the greatest integer smaller than or equal to $x$. For example $\lfloor 2.3 \rfloor = 2$ and $\lfloor -5 \rfloor = -5$. This is called the greatest integer function or the floor function.

Which of the following functions is largest for large positive values of $x$

$\begin{array}{c}&5x^2-2x+2, &4x^2-4x+3, &8x+7? \end{array}$

If $x$ goes to infinity, which of the following is the largest of all:

$\begin{array}{c}&\sqrt[x]{6^x+2^x}, &\sqrt[x]{5^x+4^x}, &\sqrt[x]{7^x}? \end{array}$

If $x$ goes to infinity, which of the following is the largest of all:

$\begin{array}{c}&x\sqrt[x]{2}, &4x, &x\sin \frac{2}{x}? \end{array}$

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