Chapter 4.6, Problem 51E

Solution

To calculate : the behaviour of the function f.

The end behaviour as $x\to \infty$ the $f(x)\to 0$ .

Given information: Function $f(x)=x^{-1}\sin 3x$ .

Formula used: linear calculation.

Calculation: The graph of $f(x)\sin bx$ oscillates between the graph of $y=f(x)$ and $y=-f(x)$ . When this reduces the amplitude of the wave then it is called damped oscillation. The factor $f(x)$ is known as the damping factor.

The graph of the function $y=x^{-1}\sin 3x$ , with the plus and minus damping factors $y=x^{-x}$ and $y=-x^{-x}$ is

  

From the graph above the function exhibits damped oscillation.

The end behaviour as $x\to \infty$ the $f(x)\to 0$