Short and Sweet Calculus

2.4Limits 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 $$\infty$$.