Chapter 11.3, Problem 54E

Solution

(a)

To find: The value of the limit $\underset{x\to \infty }{{\displaystyle \lim y}}$ for the given function $y=e^{-x}\cos x$ .

The value of the limit is $\underset{x\to \infty }{{\displaystyle \lim y}}=0$ .

Given information:

The given function is $y=e^{-x}\cos x$ .

Formula used:

The notation $\underset{x\to \infty }{{\displaystyle \lim y}}=l$ means that $y$ gets arbitrarily close to $l$ as $x$ gets arbitrarily large.

Product rule of limits:

If $\underset{x\to c}{{\displaystyle \lim f\left(x\right)}}$ and $\underset{x\to c}{{\displaystyle \lim g\left(x\right)}}$ both exist, then $\underset{x\to c}{{\displaystyle \lim }}\left[f\left(x\right)\cdot g\left(x\right)\right]=\underset{x\to c}{{\displaystyle \lim }}f\left(x\right)\cdot \underset{x\to c}{{\displaystyle \lim }}g\left(x\right)$ .

Calculation:

The given function is: $y=e^{-x}\cos x$ .

Since, $-1\le \cos x\le 1$ ,

  $\cos x$ oscillates between $-1$ and $1$ .

But,

  $e^{-x}$ tends to 0 as $x$ increases arbitrarily in the negative direction.

According to the product rule of limits,

  $\underset{x\to \infty }{{\displaystyle \lim y}}=0$ .

The graph of $y=e^{-x}\cos x$

confirms this:

  

Figure 1

Therefore, the required value of the limit is $0$ .

(b)

To find: The value of the limit $\underset{x\to -\infty }{{\displaystyle \lim y}}$ for the given function $y=e^{-x}\cos x$ .

The value of the limit is $\underset{x\to -\infty }{{\displaystyle \lim y}}=undefined$ or does not exist.

Given information:

The given function is $y=e^{-x}\cos x$ .

Formula used:

The notation $\underset{x\to \infty }{{\displaystyle \lim y}}=l$ means that $y$ gets arbitrarily close to $l$ as $x$ gets arbitrarily large.

Product rule of limits:

If $\underset{x\to c}{{\displaystyle \lim f\left(x\right)}}$ and $\underset{x\to c}{{\displaystyle \lim g\left(x\right)}}$ both exist, then $\underset{x\to c}{{\displaystyle \lim }}\left[f\left(x\right)\cdot g\left(x\right)\right]=\underset{x\to c}{{\displaystyle \lim }}f\left(x\right)\cdot \underset{x\to c}{{\displaystyle \lim }}g\left(x\right)$ .

Calculation:

The given function is: $y=e^{-x}\cos x$

  $e^{-x}$ approaches $\infty$ as $x$ approaches $\infty$

  $e^{-x}$ is then multiplied by $\cos x$ that oscillates between -1 and 1.

Therefore, $\underset{x\to -\infty }{{\displaystyle \lim y}}=undefined$

Therefore, the required value of the limit is undefined/does not exist.