Video Transcript
Assume that the limit as 𝑥 tends
to six of 𝑓 of 𝑥 is equal to three and the limit as 𝑥 tends to six of 𝑔 of 𝑥 is
equal to eight. Find the limit as 𝑥 tends to six
of the square root of 𝑔 of 𝑥 minus 𝑓 of 𝑥.
We need to find the limit as 𝑥
tends to six of the square root of 𝑔 of 𝑥 minus 𝑓 of 𝑥. We can break this limit down using
the properties of limits. We have the property for the limits
of roots of functions. It tells us that the limit as 𝑥
tends to some constant 𝑎 of the 𝑛th root of some function 𝑓 of 𝑥 is equal to the
𝑛th root of the limit as 𝑥 tends to 𝑎 of 𝑓 of 𝑥. The limit we’re trying to find is
the limit as 𝑥 tends to six of the square root of 𝑔 of 𝑥 minus 𝑓 of 𝑥. So we have the limit of a square
root of a function. We can therefore apply our rule for
limits of roots of functions. It tells us that our limit is equal
to the square root of the limit as 𝑥 tends to six of 𝑔 of 𝑥 minus 𝑓 of 𝑥.
Now, we can see that we have the
limit of a difference of functions since our limit is of 𝑔 of 𝑥 minus 𝑓 of
𝑥. We can apply the rule for the limit
of differences of functions, which tells us that the limit as 𝑥 tends to some
constant 𝑎 of a difference of functions — so that’s 𝑓 of 𝑥 minus 𝑔 of 𝑥 — is
equal to the limit as 𝑥 tends to 𝑎 of 𝑓 of 𝑥 minus the limit as 𝑥 tends to 𝑎
of 𝑔 of 𝑥. We can apply this rule to our limit
within the square root, giving us that our limit is equal to the square root of the
limit as 𝑥 tends to six of 𝑔 of 𝑥 minus the limit as 𝑥 tends to six of 𝑓 of
𝑥.
Now, we can spot that the limits
within our square root have been given to us in the question. We have that the limit as 𝑥 tends
to six of 𝑓 of 𝑥 is equal to three and the limit as 𝑥 tends to six of 𝑔 of 𝑥 is
equal to eight, giving us that our limit is equal to the square root of eight minus
three. Simplifying this, we obtain our
solution which is that the limit as 𝑥 tends to six of the square root of 𝑔 of 𝑥
minus 𝑓 of 𝑥 is equal to the square root of five.
