Calculus

Curve Sketching

Relative Magnitude of Functions

         

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

x33x,3sin1x,cos1x?\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 xx grows to infinity?

4x2+9x,2x\begin{array}{c}&\sqrt{\lfloor 4x^2+9x \rfloor}, &2x \end{array}

Details and assumptions

The function x:RZ\lfloor x \rfloor: \mathbb{R} \rightarrow \mathbb{Z} refers to the greatest integer smaller than or equal to xx. For example 2.3=2\lfloor 2.3 \rfloor = 2 and 5=5\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 xx

5x22x+2,4x24x+3,8x+7?\begin{array}{c}&5x^2-2x+2, &4x^2-4x+3, &8x+7? \end{array}

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

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

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

x2x,4x,xsin2x?\begin{array}{c}&x\sqrt[x]{2}, &4x, &x\sin \frac{2}{x}? \end{array}

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