Integral of $\frac{4}{x+1}$ from $4$ to $\infty$.

Question

I want to calculate the integral

$$\int_4^\infty\frac{4}{x+1}dx.$$

I know that the result is

$$\lim_{x\to\infty}(4 \ln (x + 1)- 4 \ln (5)),$$

then I get $\infty - \ln (625)$. Is it still infinity?

Answer 1

Short answer: yes.

Longer answer:

You do not "get" $\infty-\ln(625)$. Infinity is not a real number, and "infinite limits" do not work that way. The definition of a limit being infinite is the following:

For a function $f(x)$, the limit $$\lim_{x\to \infty} f(x)$$ is "equal" to $\infty$ if, for every $M\in \mathbb R$, there exists such a $x_0\in \mathbb R$ that for all $x>x_0$, we have $f(x) > M$.

Note that the limit is therefore "equal" to $\infty$ only if the limit does not exists, i.e. it diverges. Therefore, saying that the limit of $f$ is $\infty$ as $x\to\infty$ is really saying "no, there is no limit, but there is still some sort of structure in the behavior as $x$ becomes large".

That said, in your case, the limit of $4\ln(x+1)$ is indeed $\infty$, and because $\4\ln(x+1)$ exceeds all bounds $M$, so does $4\ln(x+1)-C$ for any constant $C$.