Video: Finding the Limit of Rational Functions at Infinity

Find lim_(π‘₯ β†’ ∞) 6π‘₯Β²/(π‘₯ βˆ’ 6).


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 𝑦 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 it’s 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; that’s 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 we’re dealing with limits, it’s perfectly fine to say that infinity plus 36 is just infinity, which gives another path to this answer.

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