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You don't need a calculator or computer to draw your graphs! Derivatives and other Calculus techniques give direct insights into the geometric behavior of curves.

If \(x\) goes to infinity, which of the following is the largest of all:

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

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Which of the following is larger in the limit as \(x \) grows to infinity?

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

**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.

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Which of the following functions is largest for large positive values of \(x\)

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

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If \(x\) goes to infinity, which of the following is the largest of all:

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

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If \(x\) goes to infinity, which of the following is the largest of all:

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

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