### Video Transcript

Determine the sum of the limit of
π of π₯ as π₯ approaches negative five, the limit of π of π₯ as π₯ approaches
negative seven from the right, and the limit of π of π₯ as π₯ approaches zero from
the left.

We are given the graph of the
function π of π₯. We will use this graph to find the
three limits needed to evaluate the sum. From the graph, we can see that π
is a piecewise defined function. It is defined by two
subfunctions. One corresponding to the interval
of π₯ values greater than or equal to negative seven but strictly less than zero and
one corresponding to the interval of π₯ values strictly greater than zero but less
than or equal to two. We are not given the algebraic
expressions for these subfunctions, but just their graphs. Note that the small hollow circles
on the π¦-axis at the points π¦ equals three and π¦ equals negative four indicate
that the function π is not defined at the π₯-value zero.

Letβs start by evaluating the limit
of π of π₯ as π₯ approaches negative five. We know that if the limit of π of
π₯ as π₯ approaches negative five exists, then it must be equal to the limit of π
of π₯ as π₯ approaches negative five from the left and the limit of π of π₯ as π₯
approaches negative five from the right. From the graph of the subfunction
corresponding to the values of π₯ greater than or equal to negative seven, but less
than zero, we see that as π₯ approaches negative five from the left, i.e., takes
values near negative five, but strictly less than negative five. And as π₯ approaches negative five
from the right, i.e., takes values near negative five, but strictly greater than
negative five. The function π of π₯ approaches
the value negative five. So the left and right limits of the
function π of π₯ as π₯ approaches negative five are both equal to negative
five. So the limit of π of π₯ as π₯
approaches negative five is also equal to negative five.

Now, letβs evaluate the limit of π
of π₯ as π₯ approaches negative seven from the right. From the graph of the subfunction
corresponding to the values of π₯ greater than or equal to negative seven but less
than zero, we see that as π₯ approaches negative seven from the right, i.e., takes
values near negative seven, but strictly greater than negative seven. The function π of π₯ approaches
the value negative one. So the limit of π of π₯ as π₯
approaches negative seven from the right is equal to negative one. Finally, letβs evaluate the limit
of π of π₯ as π₯ approaches zero from the left. Note that π₯ approaches zero from
the left means that we are considering values of π₯ near zero but strictly less than
zero. Therefore, we will consider the
subfunction corresponding to the values of π₯ greater than or equal to negative
seven but strictly less than zero when evaluating this limit.

As π₯ approaches zero from the
left, i.e., takes the values near zero but strictly less than zero, the function π
of π₯ approaches the value negative four. So the limit of π of π₯ as π₯
approaches zero from the left is equal to negative four. At this point, it might be worth
noting that as π₯ approaches zero from the right, i.e., takes the values near zero
but strictly greater than zero, the function π of π₯ approaches the value
three. So the limit of π of π₯ as π₯
approaches zero from the right is equal to three. However, this is not needed to
answer the question. In order to answer the question, we
need to add together the values negative five, negative one, and negative four. Doing so, we obtain the value
negative 10, which is our final answer.