Question

Not sure if I got the right answer and what to make sure I did it correctly. I got that the limit doesn't exist.

Answer 1

We know that the limit

$\lim_{x\to a}f(x)$

exists if and only if:

$\lim_{x\to a^-}f(x)=\lim_{x\to a^+}f(x)$

Let's calculate the left sided limit, for this limit we will use the first expression:

$\lim_{x\to8^-}f(x)=\lim_{x\to8^-}(x+10)=18$

Let's find the right sided limit, for this limit we will use the second expression:

$\lim_{x\to8^+}f(x)=\lim_{x\to8^+}(10-x)=2$

We notice that these limits are not equal which means that the limit does not exist.

We can visualize this with the graph of the function which is shown below:

From the graph we notice that as we approach eight the value of the function is not the same for each side which means that the limit does not exist.