First, we note that the function is defined at
with
, so (i) and (iii) are not correct.
Now let’s see whether the limit of the function at
exists. For this purpose, we consider the left limit and the right limit. When
approaches to
from the right, we consider points near
which is greater than
. For such points, the corresponding piece of the function is
, so that
![{\displaystyle \lim _{x\rightarrow 0^{+}}f(x)=\lim _{x\rightarrow 0^{+}}k(x-1)+2=-k+2=2}](https://wiki.ubc.ca/api/rest_v1/media/math/render/svg/e7e838a82e90f296061e49788672d4aa2e66353d)
Here we use the given information
![{\displaystyle k=0}](https://wiki.ubc.ca/api/rest_v1/media/math/render/svg/6307c8a99dad7d0bcb712352ae0a748bd99a038b)
.
On the other hand, to get the left limit, we consider points
close to
. Since the corresponding piece of the function in such region is
, we get
![{\displaystyle \lim _{x\rightarrow 0^{-}}f(x)=\lim _{x\rightarrow 0^{-}}e^{x}=e^{0}=1.}](https://wiki.ubc.ca/api/rest_v1/media/math/render/svg/8b7ed8644c15a2e2abc6041ada2409601459309b)
Since the left limit is not equal to right limit at
,
![{\displaystyle \lim _{x\rightarrow 0^{+}}f(x)\neq \lim _{x\rightarrow 0^{-}}f(x).}](https://wiki.ubc.ca/api/rest_v1/media/math/render/svg/71aebe310085c8fa01d53181fe469995e0bf27ae)
the limit
![{\textstyle \lim _{x\rightarrow 0}f(x)}](https://wiki.ubc.ca/api/rest_v1/media/math/render/svg/902e5a63a34fd49d7cad8d92642622169a5b3eab)
does not exist. In all, we choose
![{\textstyle \color {blue}{\text{(ii)}}}](https://wiki.ubc.ca/api/rest_v1/media/math/render/svg/79fa192de8d6e63d8c24716d1156bb54e825c220)
.