How Do You Calculate a Limit at Infinity?

You may have notice that there are two ways to use algebra to compute limits at infinity. Let’s look at two possible strategies for computing a limit at infinity for a rational function, \underset{x \to - \infty }{\mathop{lim }},\frac{2{{x}^{3}}+x}{3{{x}^{2}}+1}.

In the board above, I divided each term in the rational expression by x to the highest power that appears anywhere in the expression. The denominator get very small as the top approaches 2, so the entire expression must grow very big. Since the bottom is also negative, the expression grows big and is negative. Now let’s change that strategy a bit.

In the board above, I divide by x to the largest power that appears in the denominator. The bottom then gets closer and closer to 3. But the top contains 2x which grow more and more negative as x approaches - \infty. So the entire expression grows more and more negative.

Either strategy gives the correct answer as long as you interpret what is going on in the numerator and denominator correctly.