## 2.4 Limits at Infinity

Consider the function \(f\) defined by the equation \[f(x)=\frac{x+1}{x+2}.\] Let’s investigate the behavior of \(f\) when \(x\) is positive and becomes larger and larger. From the following table and the graph of \(f\) (Figure [fig:ch4-limit-at-inf-1]), we see that \(f(x)\) gets closer and closer to 1. In other words, we can make \(f(x)\) arbitrarily close to 1 if we choose \(x\) sufficiently large. In this case, we say \(f\) approaches 1 (or \(f\) has limit 1) as \(x\) approaches infinity and we write \[\lim_{x\to\infty}f(x)=1.\]

\(x\) | 10 | 100 | 1000 | 10,000 | 100,000 | 1,000,000 | … |
---|---|---|---|---|---|---|---|

\(f(x)\) | 0.916667 | 0.990196 | 0.99002 | 0.9999 | 0.99999 | 0.999999 | … |

Now let’s investigate the behavior of \(f\) when \(x\) is negative and its magnitude becomes larger and larger. In this case, we see from the following table and the graph of \(f\) that \(f(x)\) gets closer and closer to 1 too. In this case, we say \(f\) approaches 1 (or \(f\) has limit 1) as \(x\) approaches minus infinity and write \[\lim_{x\to-\infty}f(x)=1.\] [Here is a coincidence that \(\lim_{x\to+\infty}f(x)=\lim_{x\to-\infty}f(x)=1\)].

\(x\) | … | –1,000,000 | –100,000 | –10,000 | –1000 | –100 | –10 |
---|---|---|---|---|---|---|---|

\(f(x)\) | … | 1.000001 | 1.00001 | 1.00010002 | 1.001002 | 1.010204 | 0.818182 |

In general, if the graph of \(f\) gets closer and closer to the horizontal line \(y=L\) as \(x\) gets larger and larger, we say the limit of \(f\)as \(x\) approaches \(+\infty\) is \(L\) and write \[\lim_{x\to+\infty}f(x)=L.\] In a similar fashion, we can define \(\lim_{x\to-\infty}f(x)=L\).

Instead of \(+\infty\) we may simply write \(\infty\).

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