Chapter 11.3, Problem 49E

Solution

(a)

To find: The value of the limit $\underset{x\to \infty }{{\displaystyle \lim y}}$ for the given function $y=1+2^x$ .

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

Given information:

The given function is $y=1+2^x$ .

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.

Sum 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 $\underset{x\to c}{{\displaystyle \lim }}\left[f\left(x\right)+g\left(x\right)\right]=\underset{x\to c}{{\displaystyle \lim }}f\left(x\right)+\underset{x\to c}{{\displaystyle \lim }}g\left(x\right)$ .

Calculation:

The given function is: $y=1+2^x$

Substitute $x=\infty$

  $2^{\infty }$ means $2\times 2\times 2\times \mathrm{2......}$ up to infinity which tends to infinity.

i.e. $\underset{x\to \infty }{{\displaystyle \lim 2^x=\infty }}$

Calculate the limits:

  $\underset{x\to \infty }{{\displaystyle \lim y}}=\underset{x\to \infty }{{\displaystyle \lim }}\left(1+2^x\right)$

According to Sum rule of limits:

  $\underset{}{{\displaystyle \underset{x\to \infty }{{\displaystyle \lim y}}=\underset{x\to \infty }{{\displaystyle \lim 1}}+\underset{x\to \infty }{{\displaystyle \lim 2^x}}}}$

Substitute $\underset{x\to \infty }{{\displaystyle \lim 2^x=\infty }}$ and $\underset{x\to \infty }{{\displaystyle \lim 1=1}}$

in the above equation to obtain the required value of the limit:

  $\underset{}{{\displaystyle \begin{array}{l}\underset{x\to \infty }{{\displaystyle \lim y}}=1+\infty \\ \underset{x\to \infty }{{\displaystyle \lim y}}=\infty \end{array}}}$

Therefore, the required value of the limit is $\infty$ .

(b)

To find: The value of the limit $\underset{x\to -\infty }{{\displaystyle \lim y}}$ for the given function $y=1+2^x$ .

The value of the limit is $\underset{x\to -\infty }{{\displaystyle \lim y}}=1$ .

Given information:

The given function is $y=1+2^x$ .

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.

Sum 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 $\underset{x\to c}{{\displaystyle \lim }}\left[f\left(x\right)+g\left(x\right)\right]=\underset{x\to c}{{\displaystyle \lim }}f\left(x\right)+\underset{x\to c}{{\displaystyle \lim }}g\left(x\right)$ .

Calculation:

The given function is: $y=1+2^x$

Substitute $x=-\infty$

  $2^{-\infty }$ means $\frac{1}{2}\times \frac{1}{2}\times \frac{1}{2}\times \frac{1}{2}\times \mathrm{......}$ up to infinity which tends to $0$ .

i.e. $\underset{x\to -\infty }{{\displaystyle \lim 2^x=0}}$

Calculate the limits:

  $\underset{x\to -\infty }{{\displaystyle \lim y}}=\underset{x\to -\infty }{{\displaystyle \lim }}\left(1+2^x\right)$

According to Sum rule of limits:

  $\underset{}{{\displaystyle \underset{x\to -\infty }{{\displaystyle \lim y}}=\underset{x\to -\infty }{{\displaystyle \lim 1}}+\underset{x\to -\infty }{{\displaystyle \lim 2^x}}}}$

Substitute $\underset{x\to -\infty }{{\displaystyle \lim 2^x=0}}$ and $\underset{x\to -\infty }{{\displaystyle \lim 1=1}}$

in the above equation to obtain the required value of the limit:

  $\underset{}{{\displaystyle \begin{array}{l}\underset{x\to -\infty }{{\displaystyle \lim y}}=1+0\\ \underset{x\to -\infty }{{\displaystyle \lim y}}=1\end{array}}}$

Therefore, the required value of the limit is $1$ .