Given:
There are given that the function:
[tex]f(x)=\frac{1}{(x+3)}-2[/tex]
Explanation:
The graph of the given function is shown below:
Now,
(1) Domain:
To find the domain of the given function, we need to find the value where the function is defined.
Then,
The domain of the given function is:
[tex]\text{Domain: (-}\infty,-3)\cup(-3,\infty)[/tex]
(2) Range:
To find the range of the given function, we need to find the set of values that correspond with the domain.
So,
The range of the given function is:
[tex](-\infty,-2)\cup(-2,\infty)[/tex]
(3) Increasing on:
[tex]\text{ increasing on: never increasing}[/tex]
(4) Decreasing on:
The value of decreasing on:
[tex](-\infty,-3),(-3,\infty)[/tex]
(5): All asymptote:
The value of asymptote are:
[tex]\begin{gathered} \text{vertical asymptote : x=-3} \\ \text{Horizontal asymptote : y=-2} \end{gathered}[/tex]
(6) All limit (4):
[tex]\begin{gathered} f(x)=\frac{1}{(x+3)}-2 \\ f(4)=\frac{1}{(4+3)}-2 \\ f(4)=\frac{1}{7}-2 \\ f(4)=\frac{-13}{7} \end{gathered}[/tex]
Hence, the all limit at 4 is -1.85.
