### Video Transcript

Determine the left-sided limit as
π₯ approaches negative nine of π of π₯ and the right-sided limit as π₯ approaches
negative nine of π of π₯ given that π of π₯ is equal to π₯ plus nine if π₯ is less
than or equal to negative nine and one over π₯ plus nine if π₯ is greater than
negative nine.

Here, weβve been given a function
defined piecewise over two intervals. For the left-sided limit, weβre
approaching π₯ equals negative nine from the negative direction. Hence, π₯ is less than negative
nine. For the right-sided limit, weβre
approaching π₯ equals negative nine from the positive direction. And hence, π₯ is greater than
negative nine. Since π₯ equals negative nine is
the point between the two intervals of our piecewise function, for our left-sided
limit will be in the first interval and for our right-sided limit will be in the
second interval. Letβs work on finding the
left-sided limit.

In this case, our function π of π₯
is π₯ plus nine. We can find this limit by direct
substitution of π₯ equals negative nine into our function. Doing so, we find that our answer
is negative nine plus nine which is equal to zero. The left-sided limit as π₯
approaches negative nine of π of π₯ is therefore zero. Now for the right-sided limit, here
our function π of π₯ is one over π₯ plus nine. Again, we tried direct substitution
of π₯ equals negative nine into our function. This time, doing so gives us an
answer of one over zero. And as we know, dividing one by
zero cannot be evaluated to a numerical value. In cases such as this, we say that
the limit does not exist. And so in a strict sense, this is
the answer to our question.

To more fully understand our
result, however, letβs look at a graphical plot of our function. Here, we have sketched our
graph. And we know that, in the interval
where π₯ is less than or equal to negative nine, we have a well behaved
function. And we know that due to the solid
dot here at negative nine π₯ is indeed defined at this point. For the other interval, we know
that as π₯ approaches negative nine, we have a vertical asymptote. This means that the values of π of
π₯ get arbitrarily large. And this is often represented as
infinity. In this sense, it is common to
write that the right-sided limit as π₯ approaches negative nine of π of π₯ is equal
to positive infinity.