### Video Transcript

Discuss the existence of the limit
as π₯ approaches seven of π of π₯ given π of π₯ is equal to 13π₯ plus seven if one
is less than π₯ which is less than seven and 14π₯ plus seven if seven is less than
or equal to π₯ which is less than eight.

In this example, our π of π₯ is a
piecewise function. And we are asked to find the limit
as π₯ approaches seven. Seven is the π₯ value at which our
function switches between 13π₯ plus seven and 14π₯ plus seven. In order to find whether our limit
exists, we need to check whether the left and right limits exist and if they are
equal. Weβll start by considering the left
limit. Since π₯ is approaching seven from
below, we know that π₯ must be less than seven. Since π₯ is less than seven, we can
see from our piecewise definition that π of π₯ is equal to 13π₯ plus seven.

Since this is a polynomial, we can
use direct substitution. In order to find this limit, we
simply substitute π₯ equals seven into 13π₯ plus seven. And this gives us that the left
limit is equal to 98. Since the limit here is equal to a
real constant, we know that this limit must exist. Letβs now consider the limit as π₯
approaches seven from above. Since π₯ is approaching seven from
above, we have that π₯ is greater than seven. Therefore, from our piecewise
definition, we have that π of π₯ is equal to 14π₯ plus seven which is again a
polynomial. And so we can use direct
substitution in order to find this limit. Substituting π₯ equals seven into
14π₯ plus seven, we obtain the limit as π₯ approaches seven from the right is equal
to 105. Therefore, our limit exists.

Now, we have found that both the
left and right limit exist. However, the left limit is equal to
98. And the right limit is equal to
105. Therefore, we can conclude that the
limit as π₯ approaches seven of π of π₯ does not exist because the left and the
right limit are not equal to each other. In this example, we saw how the
limit did not exist because the left and right limits were not equal. This is because there is a jump in
the function at the point which weβre trying to take the limit. Therefore, we cannot say that the
limit of π of π₯ approaches a particular point since it depends upon which
direction we are approaching the limit as to what the limit could equal.