### Video Transcript

Assume that the limit as π₯ tends
to six of π of π₯ is equal to three and the limit as π₯ tends to six of π of π₯ is
equal to eight. Find the limit as π₯ tends to six
of the square root of π of π₯ minus π of π₯.

We need to find the limit as π₯
tends to six of the square root of π of π₯ minus π of π₯. 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 π₯
tends to some constant π of the πth root of some function π of π₯ is equal to the
πth root of the limit as π₯ tends to π of π of π₯. The limit weβre trying to find is
the limit as π₯ tends to six of the square root of π of π₯ minus π of π₯. 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 π₯ tends to six of π of π₯ minus π of π₯.

Now, we can see that we have the
limit of a difference of functions since our limit is of π of π₯ minus π of
π₯. We can apply the rule for the limit
of differences of functions, which tells us that the limit as π₯ tends to some
constant π of a difference of functions β so thatβs π of π₯ minus π of π₯ β is
equal to the limit as π₯ tends to π of π of π₯ minus the limit as π₯ tends to π
of π of π₯. 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 π₯ tends to six of π of π₯ minus the limit as π₯ tends to six of π of
π₯.

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 π₯ tends
to six of π of π₯ is equal to three and the limit as π₯ tends to six of π of π₯ 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 π₯ tends to six of the square root of π of π₯
minus π of π₯ is equal to the square root of five.