Evaluate $\lim_{x\to\infty}\left(1+\frac{\ln x}{f(x)}\right)^{f(x)/x}$

Question

Let's consider the function $f:\mathbb{R}\rightarrow(0,\infty)$, with $f(x)\cdot \ln f(x)=e^x$, $\forall x \in \mathbb{R}$. Then compute

$$\lim_{x\to\infty}\left(1+\frac{\ln x}{f(x)}\right)^{\dfrac{f(x)}{x}}$$

The first solution

Since $$f(x)\cdot \ln f(x)=e^x, \forall x \in \mathbb{R}$$ we may easily deduce that $$\lim_{x\to\infty}f(x)=\infty$$ On the other hand $$f^2(x)> f(x)\cdot \ln f(x)=e^x$$ $$f(x)> e^{x/2}$$ and then $$0\le\lim_{x\to\infty} \frac{\ln x}{f(x)}\le\frac{\ln x}{e^{x/2}}\rightarrow 0$$ $$\lim_{x\to\infty} \frac{\ln x}{f(x)}=0$$ $$\lim_{x\to\infty} \displaystyle\frac{e^{x/2}}{x} \le \lim_{x\to\infty} \displaystyle\frac{f(x)}{x}=\infty$$

At this point we recognize that our limit case is $1^{\infty}$, and have $$\lim_{x\to\infty}\left(1+\frac{\ln x}{f(x)}\right)^{\displaystyle\frac{f(x)}{x}}=\lim_{x\to\infty}e^{\displaystyle \frac{\ln x}{x }}=e^0=1$$

The second solution (from a brilliant friend of mine - - so sorry I missed this way)

Let's take log of both sides of the limit

$$\ln L = \lim_{x\to\infty} \frac{f(x)}{\ln x} \ln \left(1+\frac{\ln x}{f(x)}\right)\times \lim_{x\to\infty} \frac{\ln x}{x}=1\times 0=0$$ that is simply justified by the fact that $f(x)>>x$ (see $f(x)> e^{x/2}$)

Question: how would you approach this question? Thank you.

Answer 1

This is a very good point knowing that if $$\lim_{x\to{+\infty}} f(x)^{g(x)}=1^{+\infty}$$ which is indeterminate limit then we can solve it by taking the following limit: $$k =\lim_{x\to +\infty}\big(f(x)-1\big)g(x)$$ instead. So $$\lim_{x\to{+\infty}} f(x)^{g(x)}=e^k$$

Now, try to use this formula also. It gives you $~~\text{e}~~$ at last. For more see how @Brian proved me that. This proof deserves more that +100. 100

Answer 2

$\lim\limits_{x\to+\infty}\left(1+\frac{\ln x}{f(x)}\right)^{\displaystyle\frac{f(x)}{x}}$

$\left(1+\frac{x}{f(e^x)}\right)^{\displaystyle\frac{f(e^x)}{e^x}} = \left(\left(1+\frac{x}{f(e^x)}\right)^{\displaystyle f(e^x)}\right)^{\displaystyle\frac{1}{e^x}} \sim\big(e^x\big)^{\displaystyle\frac{1}{e^x}} = \big(e\big)^{\displaystyle\frac{x}{e^x}} \to e^0 = 1$

The substitution $x \leftarrow e^x$ is justified by the fact that $x \to +\infty \Leftrightarrow e^x \to +\infty$

The $\sim$ by the fact that $f(e^x) \to +\infty$ (not completely sure about that)

And the rest is simple.

Answer 3

The first step is to notice that $f$ is differentiable: Differentiating the equation for $f$ with respect to $f$ (we're going to use the inverse function theorem here, so we're thinking of $x$ as a function of $f$) and dividing through by $e^x$ we get $$\frac{dx}{df}= \frac{1+\ln{f}}{e^x}=\frac{1+\frac{e^x}{f}}{e^x}.$$ But $f$ is by definition always positive. Therefore, the $dx/df$ can never be zero, which by the inverse function theorem means that $f$ must always be differentiable. Moreover, since $dx/df$ is always positive, so is $df/dx$, which means--and here's the punch line--that $f$ is (strictly) monotone increasing.

So, now let's try and find some bounds on $f$. Since $f$ is monotone increasing, it's going to eventually be strictly positive (in fact it one can show that it'll always be positive, but that's not going to be important to us), so we can use the fact that $\ln{x}<x$ for all $x>0$ to get that $\ln{f(x)}<f(x)$ for large enough $x$. Therefore, returning to the defining equation for $f$, we get the two inequalities $$e^x=f(x)\ln{f(x)}\leq f(x)^2$$ and $$e^x\geq[\ln{f(x)}]^2.$$ These simplify to $$e^{x/2}\leq f(x) \leq e^{e^{x/2}}.$$

Finally, let's head over to our limit, the inside of which I'll call $y$. Taking the natural log, we get $$\ln{y} = \frac{f(x)}{x}\ln\left(1+\frac{\ln{x}}{f(x)}\right).$$ Using $f(x)\geq e^{x/2}$, we can get rid of the $f(x)$ outside of the $\ln$, and using $f(x)\leq e^{e^{x/2}}$, we can get rid of the one inside. This gives $$\ln{y} \leq \frac{e^{x/2}}{x} \ln\left(1+e^{-e^{x/2}}\log{x}\right),$$ which I'll leave to you to show approaches $0$ as $x\to \infty$.

We'd like to simply stop here and say that $\ln{y}$ must also approach $0$, but this only works if $\ln{y}$ is (eventually) nonnegative---then we can use the squeeze theorem to conclude that yes, $\ln{y}$ does in fact approach $0$. So, is $\ln{y}$ nonnegative? Well, let's look back at the original equation for $\ln{y}$. Since $f$ is always positive, the part outside the $\ln$ is also positive (as long as $x>0$, but we're working with large $x$ here, so we can assume this). The same reasoning tells us that the part inside the $\ln$ must be greater than $1$ (as long as $x>1$, but again, we can assume this), and hence that the $\ln$ part itself is positive. Thus, $\ln{y}$ is indeed positive, and we get that $\ln{y}\to 0$ as $x\to 0$.

Hence, $$\lim_{x\to\infty} \left(1+\frac{\ln{x}}{f(x)}\right)^{\dfrac{f(x)}{x}}=1.$$

Answer 4

Since $g(y) =y\log y$ is strictly increasing for $y>1$ it follows that there is an inverse function $h$ which is strictly increasing on $(0,\infty) $.

Now the given function $f$ is given by $f(x) =h(e^x) $ and clearly $f(x) \to\infty $ as $x\to\infty $. The given limit is easily handled by putting $t=f(x) $ so that $x=\log g(t) =\log(t\log t) $ and then the desired limit is $$\lim_{t\to\infty} \left(1+\frac{\log\log(t\log t)} {t} \right) ^{t/\log(t\log t)} $$ Since $\dfrac{\log\log(t\log t)}{t} \to 0$ it follows that the desired limit equals $$\exp\left(\lim_{t\to\infty} \frac{\log\log(t\log t)} {\log(t\log t)} \right) =e^0=1$$