Question

Please help verify if my answers are correct! Thank you:)

Answer 1

SOLUTION

From the graph of the function giving, Consider the image below

$\begin{gathered} \text{when x=-1.f(-1)=1} \\ \text{Hence } \\ f(-1)=1 \end{gathered}$

Hence

F(-1)=1

From the graph,

$\begin{gathered} \text{when }x=1,f(1) \\ \text{Hence } \\ f(1)=1 \\ \text{The unshaded point shows the number is not included } \end{gathered}$

hence

F(1)=1

For

$\begin{gathered} \text{when x=6,f(6)=0} \\ \text{Hence } \\ f(6)=0 \end{gathered}$

Hence

f(6)=0

The limit of f(x) as x approaches zero

$\begin{gathered} \text{The Right hand limit is } \\ \lim _{x\rightarrow0^+}f(x)=-1 \\ \text{and } \\ \text{The left hand limit} \\ \lim _{x\rightarrow0^-}f(x)=0 \end{gathered}$

Since the Left hand limit is not equal to the Right Linit (DLR)

The limit does not exist(DNE)

The limit of f(x) as x approaches two is

$\begin{gathered} \text{The right hand limit is } \\ \lim _{x\rightarrow2^+}f(x)=0 \\ \text{And } \\ \text{The left hand limit is } \\ \lim _{x\rightarrow2^-}f(x)=0 \end{gathered}$

The Left hand limit equalis the right hand Limit,

Hence Limit has x approaches 2 is 0

The limit of f(x) as x approaches 4 is given as

$\begin{gathered} \text{The right hand limit is } \\ \lim _{x\rightarrow4^+}f(x)=-2 \\ \text{And The right hand limit is } \\ \lim _{x\rightarrow4^-}f(x)=-2 \end{gathered}$

From the graph, the function is not continous at f(x)=-2, but the limit exist at that point since the Left hand limit and the right hand limit are the same.

Hence

Limit as x approaches 4 is -2