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Prove the following limit without using approximations and derivatives: $$\lim_\limits{x\to -\infty}{\frac{\ln\left(1+3^{x}\right)}{\ln\left(1+2^{x}\right)}}=0$$

I cannot think of any possible factorization or inequality (so that I could use the Squeeze Theorem) that doesn't use derivatives so as to find this limit. Any hint?

Harish Chandra Rajpoot
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2 Answers2

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HINT:

$$\frac{z}{z+1} \le \log (1+z)\le z \tag 1$$

for $z>-1$. Then, use the squeeze theorem.

SPOILER ALERT: Scroll over the highlighted region to reveal the solution.

Using $(1)$, we have $$\frac{3^x}{2^x(1+3^x)}<\frac{\log (1+3^x)}{\log (1+2^x)}<\frac{3^x(1+2^x)}{2^x}\tag 2$$As $x\to -\infty$ both the left-hand and right-hand sides of $(2)$ approach $0$. Therefore, by the squeeze theorem, the limit of the sequence of interest goes to $0$.

NOTE 1:

We can prove the inequalities in $(1)$ using standard tools, which don't require the use of derivatives. We need only to note that

$$e^x\ge 1+x \tag 2$$

For $x>-1$, we take the log of both sides to reveal

$$\log(1+x)\le x$$

which is the right-hand side inequality of $(1)$. Likewise, if we make the substitution $x=-z/(z+1)$ in $(2)$, we obtain

$$e^{-z/(1+z)}\ge 1-\frac z{z+1}=\frac1{z+1}$$

whereupon rearranging yields for $z>-1$

$$\log(1+z)\ge \frac{z}{z+1}$$

as was to be shown.

NOTE 2:

We can prove the inequality given in $(2)$ using the limit definition of $e^x$

$$e^x=\lim_{n\to \infty}\left(1+\frac xn\right)^n$$

I showed in THIS ANSWER, without the use of derivatives, that $\left(1+\frac xn\right)^n$ is a monotonically increasing function of $n$ (If $x<0$, this is true for $n>|x|$). Then, we have

$$e^x\ge \left(1+\frac xn\right)^n\ge 1+x$$

where this last inequality follows from Bernoulli's inequality. And we are done!

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Mark Viola
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  • (1) is not known by theory we have until now, so I cannot use it without proving it. But it is proven only with the use of derivatives, so I cannot "officially" prove it... –  Sep 22 '15 at 19:51
  • One does not need differential or integral calculus to prove those inequalities. – Mark Viola Sep 22 '15 at 20:16
  • I lost you, after "an increasing function of n". How do you conclude the following using monotony? $$e^x\geq \left(1+\frac{x}{n}\right) ^n$$ –  Sep 23 '15 at 17:44
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    @Jason You're welcome! My pleasure. And thank you for the "best vote!" Much appreciative. – Mark Viola Sep 24 '15 at 17:30
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Notice, $$\lim_{x\to -\infty}\frac{\ln(1+3^x)}{\ln(1+2^x)}$$ substituting $x=-t$, we get
$$\lim_{t\to \infty}\frac{\ln(1+3^{-t})}{\ln(1+2^{-t})}$$ $$=\lim_{t\to \infty}\left(\frac{\frac{\ln(1+3^{-t})}{3^{-t}}}{\frac{\ln(1+2^{-t})}{2^{-t}}}\right)\times \frac{2^t}{3^t}$$

$$=\left(\frac{\lim_{t\to \infty}\frac{\ln(1+3^{-t})}{3^{-t}}}{\lim_{t\to \infty}\frac{\ln(1+2^{-t})}{2^{-t}}}\right)\times\lim_{t\to \infty} \left(\frac{2}{3}\right)^t$$ $$=\frac{1}{1}\times (0)=\color{red}{0}$$

Harish Chandra Rajpoot
  • 37,450
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    How do you know the $\lim_\limits{x\to +\infty}{\frac{\ln(1+x)}{x}}$ without derivatives? I think it cannot be used... –  Sep 22 '15 at 19:49
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    @Jason $$\ln(1+x)=\int_0^x\frac{du}{1+u}\implies\frac{x}{1+x}\leqslant\ln(1+x)\leqslant x\implies\lim_{x\to0}\frac{\ln(1+x)}{x}=\cdots$$ – Did Sep 22 '15 at 21:30
  • @Did You are using integrals. We haven't even be taught "officially" derivatives, so integrals are a bit... –  Sep 23 '15 at 17:37
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    @Jason Quote: "Prove the following limit without using approximations and derivatives". – Did Sep 23 '15 at 18:55
  • @Did Those are the bounds that I used in my answer and I obtained them quickly using the identical approach as you "Did" (sorry for the superfluous pun). After the OP requested, I provided a way forward to establish those bounds without appealing to differential or integral calculus. But, I agree with you ... the quote was explicit. – Mark Viola Sep 23 '15 at 19:42
  • @Dr.MV Indeed. The substitution $x=−z/(z+1)$ is nice, I did not know that. – Did Sep 23 '15 at 19:49
  • @Did I derived it yesterday while I was working out at the gym. I was not aware of this either. In fact, I backed into it through a couple of successive substitutions. Anyway, thank you for the "nice" comment. You are one of the "strongest " members that I have encountered since joining in late February. So, a comment like that really means a lot to me! – Mark Viola Sep 23 '15 at 19:53