vagnhestagn

2022-09-01

Compute $\underset{n\to \mathrm{\infty}}{lim}\mathrm{ln}(3n+7)-\mathrm{ln}(n)$

The reason why I'm having trouble with this problem is because it involves natural log (ln) and I need to find the limit.

I need to find $\underset{n\to \mathrm{\infty}}{lim}\mathrm{ln}(3n+7)-\mathrm{ln}(n)$

I noticed that as $n$ approaches infinity, $-\mathrm{ln}(n)$ should be approaching $-\mathrm{\infty}$ but I'm having trouble finding the limit since $\mathrm{ln}(3n+7)$ is in the sequence.

The reason why I'm having trouble with this problem is because it involves natural log (ln) and I need to find the limit.

I need to find $\underset{n\to \mathrm{\infty}}{lim}\mathrm{ln}(3n+7)-\mathrm{ln}(n)$

I noticed that as $n$ approaches infinity, $-\mathrm{ln}(n)$ should be approaching $-\mathrm{\infty}$ but I'm having trouble finding the limit since $\mathrm{ln}(3n+7)$ is in the sequence.

Garrett Valenzuela

Beginner2022-09-02Added 9 answers

Hint:

$$\mathrm{ln}(x)-\mathrm{ln}(y)=\mathrm{ln}\left(\frac{x}{y}\right)$$

Edit:

$$\mathrm{ln}(3n+7)-\mathrm{ln}(n)=\mathrm{ln}\left(\frac{3n+7}{n}\right)=\mathrm{ln}(3+\frac{7}{n})$$

$$\text{If}\underset{n\to \mathrm{\infty}}{lim}\frac{7}{n}=0\text{, what remains?}$$

$$\mathrm{ln}(x)-\mathrm{ln}(y)=\mathrm{ln}\left(\frac{x}{y}\right)$$

Edit:

$$\mathrm{ln}(3n+7)-\mathrm{ln}(n)=\mathrm{ln}\left(\frac{3n+7}{n}\right)=\mathrm{ln}(3+\frac{7}{n})$$

$$\text{If}\underset{n\to \mathrm{\infty}}{lim}\frac{7}{n}=0\text{, what remains?}$$

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