### Video Transcript

Determine the limit of the function
negative π₯ squared plus 10π₯ all divided by π₯ squared minus 100 as π₯ approaches
negative 10, if it exists, by evaluating the function at the following values of
π₯. Negative 9.5, negative 9.9,
negative 9.95, negative 9.99, negative 9.999, negative 9.9999, negative 10.5,
negative 10.1, negative 10.05, negative 10.01, negative 10.001, and negative
10.0001. Rounding the result. to The nearest
six decimal places.

First, we noticed that the question
is asking us to determine the limit of this function as π₯ is approaching negative
10. Then the question asks us to do
this by evaluating this function of these values of π₯. Then if it is applicable, we will
need to round our answer to the nearest six decimal places. We know that the limit of a
function π as π₯ approaches π is equal to πΏ when both the left hand and right
hand limit exist and are both equal to πΏ. So in our question, weβre given
this value as our function π. And weβre given that π₯ is
approaching negative 10.

Now, since the question is asking
us to determine this limit by evaluation, we can use a function table. So we make a table with our input
values of π₯ on the top and our output values of π of π₯ on the bottom. Since weβre going to use this
function table to determine the limit as π₯ approaches negative 10, we will include
an π₯-value at negative 10 in our table. But we will leave the output value
blank. Next, we add in all the π₯-values
given to us in the question. We take care to write these into
our table in ascending order. So we start with negative 10, the
most negative value. And we will end with negative
9.5.

Now, we want to fill in all the
values for our outputs of π of π₯. We do this by substituting in each
value of π₯ in our table into the function π. And then we will evaluate the
output. So when π₯ is negative 10.5, weβll
have the functionβs output of negative 10.5 is equal to negative 10.5 all squared,
multiplied by minus one plus 10 multiplied by negative 10.5 all divided by negative
10.5 squared minus 100. We can evaluate this to give us
negative 215.25 divided by 10.25. We can then evaluate this as
negative 21. This is the output of the function
π when π₯ is equal to negative 10.5. Therefore, we can fill in this
value of negative 21 into our function table, here.

We can now do the same with the
next value in our table, negative 10.1. We get this expression for the
output of our function π when π₯ is equal to negative 10.1. We can then evaluate this
expression. And we find that it gives us a
value of negative 101. And now, just like before, we can
fill in this value of negative 101 into our function table, here. We can then follow the same process
for every single π₯-value given in the question which will give us the following
table. Now, since our definition requires
that the limit from both the left-hand side and the right-hand side exist and are
equal, we want to evaluate the left-hand limit and the right-hand limit, using our
table. So we can evaluate the left-hand
side limit by looking at these values in our table.

If we were to write these output
values in a sequence, we would get the sequence negative 21, negative 101, negative
201, negative 1001, negative 10001, and then negative 100001. We can see that the sequence is
getting progressively more negative. And therefore it does not
converge. In fact, if we look at the outputs
close to negative 10, we see that just a tiny increase in our input because the
output decreased from around negative 10000 to around negative 100000. Again, this indicates that our
limit from the left does not converge. We can use the same methods to
analyze the right-hand limit by looking at these values in our table. We can write the outputs in the
following sequence where we noticed that the respective inputs of the sequence are
approaching negative 10.

We can see that each value in the
sequence is getting larger and larger and does not seem to be converging to any
value. This would indicate to us that the
right-hand limit as π₯ approaches negative 10 does not exist. In fact, if we look at the outputs
close to negative 10, we see that just a tiny decrease in our input equals the
output increase from around 10000 to around 100000. Again, this indicates that our
limit from the right does not converge. Since the output values from the
left and right are not approaching the same value, we can conclude that the limit in
our question does not exist.