### Video Transcript

Find the limit of six π₯ squared
over π₯ minus six as π₯ approaches infinity.

There are various ways to find this
limit. One way is to look at the graph of
π¦ equals six π₯ squared over π₯ minus six. It looks like, as π₯ increases
without bound, six π₯ squared over π₯ minus six also increases without bound. As a result, we can say that this
limit is infinity. But you might not be convinced by
this. Perhaps, the graph does something
slightly different further along the π₯-axis.

We can also perform a polynomial
long division to find out six π₯ squared over π₯ minus six equals six π₯ plus 36
plus 216 over π₯ minus six. And itβs straightforward to take
limits on the right-hand side. We can do this term-by-term. The limit of six π₯, as π₯
approaches infinity, must be infinity. As π₯ increases without bound, six
π₯ also increases without bound. The limit of 36, as π₯ approaches
infinity, is just 36. This is the limit of a
constant.

And the last limit might be a bit
more tricky. We divide numerator and denominator
by the highest power of π₯ that we see; thatβs π₯. The limit of a quotient is the
quotient of the limits. And the limit in the numerator is
just zero and in the denominator is just one. So, the value of this limit is
zero. So, our limit is infinity plus
36. And when weβre dealing with limits,
itβs perfectly fine to say that infinity plus 36 is just infinity, which gives
another path to this answer.