Chapter 12, Problem 1PS

a.

To determine

Verify the graphs could be the graph of 1y1y3 $f$

  $\underset{x\to 2}{\lim }f\left(x\right)=3$

Answer to Problem 1PS

  $\boxed{{\text{g}}_1{\text{\&g}}_4}$

Explanation of Solution

Given information:

Consider the graphs of the four functions $g_1,g_2,g_3,$ and $g_4$ .

  

  

For each given condition of the function $f$ which of the graphs could be the graph of $f$ ?

  $\underset{x\to 2}{\lim }f\left(x\right)=3$

Calculation:

Consider the graph of following functions $g_1,g_2,g_3,$ and $g_4$ .

  

  

  

  

To identify the graphs that satisfy the conditions, proceed as follows.

Observe the behaviour of graph as $x$ approaches $0$

  $\begin{array}{cccc}Function& At,x=2& Left-of,x=2& Right-of,x=2\\ g_1& 3& 3& 3\\ g_2& 2& 2& 2\\ g_3& 3& 3& 1\\ g_4& 3& 3& 3\end{array}$

Now consider the statement:

  $\underset{x\to 2}{\lim }f(x)=3$

To identify the graphs that satisfy the given conditions, proceed as follows.

Identify the graphs that has limit of the function to either side of $2$ as $3$ and also the functional value at $x=2as3$ .

On observing from the table, functions $g_1$ and $g_4$ satisfy the given conditions.

Hence, the functions are $\boxed{{\text{g}}_1{\text{\&g}}_4}$

b.

To determine

Find the graphs could be the graph of $f$

  $\underset{x\to 2^{-}}{\lim }f\left(x\right)=3$

Answer to Problem 1PS

  $\boxed{ g_1,g_3\&g_4 }$

Explanation of Solution

Given information:

Consider the graphs of the four functions $g_1,g_2,g_3,$ and $g_4$ .

  

  

For each given condition of the function $f$ which of the graphs could be the graph of $f$ ?

  $\underset{x\to 2^{-}}{\lim }f\left(x\right)=3$

Calculation:

Consider the graph of following functions $g_1,g_2,g_3,$ and $g_4$ .

  

  

  

  

To identify the graphs that satisfy the conditions, proceed as follows.

Observe the behaviour of graph as $x$ approaches $0$

  $\begin{array}{cccc}Function& At,x=2& Left-of,x=2& Right-of,x=2\\ g_1& 3& 3& 3\\ g_2& 2& 2& 2\\ g_3& 3& 3& 1\\ g_4& 3& 3& 3\end{array}$

Now consider the statement:

  $\underset{x\to 2^{-}}{\lim }f\left(x\right)=3$

To identify the graphs that satisfy the given conditions, proceed as follows.

Identify the graphs that has limit of the function to left of $2$ as $3$

On observing from the table, functions $g_1$ , $g_3$ and $g_4$ satisfy the given conditions.

Hence, the functions are $\boxed{ g_1,g_3\&g_4 }$

c.

To determine

Find the graphs could be the graph of $f$

  $\underset{x\to 2^{+}}{\lim }f\left(x\right)=3$

Answer to Problem 1PS

  $\boxed{ g_1\&g_4 }$

Explanation of Solution

Given information:

Consider the graphs of the four functions $g_1,g_2,g_3,$ and $g_4$ .

  

  

For each given condition of the function $f$ which of the graphs could be the graph of $f$ ?

  $\underset{x\to 2^{+}}{\lim }f\left(x\right)=3$

Calculation:

Consider the graph of following functions $g_1,g_2,g_3,$ and $g_4$ .

  

  

  

  

To identify the graphs that satisfy the conditions, proceed as follows.

Observe the behaviour of graph as $x$ approaches $0$

  $\begin{array}{cccc}Function& At,x=2& Left-of,x=2& Right-of,x=2\\ g_1& 3& 3& 3\\ g_2& 2& 2& 2\\ g_3& 3& 3& 1\\ g_4& 3& 3& 3\end{array}$

Now consider the statement:

  $\underset{x\to 2^{+}}{\lim }f\left(x\right)=3$

To identify the graphs that satisfy the given conditions, proceed as follows.

Identify the graphs that has limit of the function to right of $2$ as $3$

On observing from the table, functions $g_1$ , and $g_4$ satisfy the given conditions.

Hence, the functions are $\boxed{ g_1\&g_4 }$