Solving $\lim_{n\to\infty}(n\int_0^{\pi/4}(\tan x)^ndx)$? $$f(x)=\lim_{n\to\infty}\biggl(n\int_0^{\pi/4}(\tan x)^n\,dx\biggr)$$
I try to this way, $\tan x\ge x$, when $x\in(0,\frac\pi4)$, but this turns out to be $\tan x\ge0$, which is obvious even without calculation.
I think it can be solved by using the squeeze rule but I can't find a proper scaler to meet $A=g(x)\le f(x)\le h(x)=A$ when $n\to\infty$ 
$\color{red}{Edit}$:
According to@Khallil's advice,
I can solve that $f(n)+f(n+2)=\frac1{n+1}$, and it's trivial that $\frac1{n+1}=f(n)+f(n+2)\le 2f(n)\le f(n-2)+f(n)=\frac1{n-1}$.
So $f(x)=\frac12$.
This is easy and quick. Btw, is there any other idea?
$\color{red}{Edit[2]}$:
I find that @Byron Schmuland 's method is especially useful in a specific kind of problem. $$f(x)=\lim_{n\to\infty}\int_0^1\frac{x^n}{1+x}dx$$,for example, Let $X_,...,X_n$ be i.i.d. random variables with density$f(x)=1$ on $(0,1)$, CDF(cumulative density function) of $X$ is $F(x)=x$. Now let $M=\max(X_1,\dots,X_n)$, its density function is:
$f_M(x)=nx^{n-1}$ for $x\in(0,1)$
Also, it's also not hard to see that $M\to1$ in distribution as $n\to\infty$.
So,$$f(x)=\lim_{n\to\infty}\int_0^1\frac{x^n}{1+x}dx=\lim_{n\to\infty}\frac1n\int_0^1{f_M(x)\frac{x}{x+1}dx}=\lim_{n\to\infty}\frac1nE(\frac{M}{M+1})=0$$
Although this simple example can be solved by other ways more easily, but this gives us another perspective.
 A: Here's another idea, like you asked:
$$ \lim_{n \to \infty} n\int_0^{\pi/4} (\tan x)^n dx =  \lim_{n \to \infty} \int_0^1  \frac{n u^n}{u^2+1} du $$ using $ u= tanx $ and then
$$  \lim_{n \to \infty} \int_0^1  \frac{n u^n}{u^2+1} du = \lim_{n \to \infty } \int_0^1 \frac{y^{\frac{1}{n}}}{1+y^{\frac{2}{n}}} dy  $$ using $ y= u^n$ 
Note that, in this form, we have "absorbed" the $n$ that tends to infinity in a way that we can easily see what happens in the limit. Namely, the exponents tend to zero, so $y$ to the exponent tends to $1$ and hence 
$$ \lim_{n \to \infty } \int_0^1 \frac{y^{\frac{1}{n}}}{1+y^{\frac{2}{n}}} dy = \int_0^1 \frac{dy}{1+1} = \frac{1}{2} \int_0^1 dy = \frac{1}{2} $$ and you obtain your result after an easy integral.
A: Here's a solution based on order statistics, similar to my answer here.
Let $X_1,\dots, X_n$ be i.i.d. random variables with density $f(x)=1+\tan(x)^2$ on $(0,\pi/4)$. 
The distribution function of $X$ is $F(x)=\tan(x)$ for $0\leq x\leq \pi/4$.
Now let $M=\max(X_1,\dots, X_n)$; its density function is 
$$f_M(x)=n F(x)^{n-1}f_X(x)=n\,(\tan(x))^{n-1}\,(1+\tan(x)^2)\text{ for }0\leq x\leq \pi/4.$$ 
Also, it is not hard to see that  $M\to \pi/4$ in distribution as $n\to\infty$.
Now $$\int_0^{\pi/4} n \tan(x)^n \,dx =\int_0^{\pi/4} {\tan(x)\over 1+\tan(x)^2}\, f_M(x) \,dx
=\mathbb{E}\left({\tan(M)\over 1+\tan(M)^2}\right).$$ 
Since $\tan(\pi/4)=1$, this converges to 
${1\over 1+1}={1\over 2}$ as $n\to\infty$.
A: After letting $\tan(x)\mapsto x$ and using the elementary limit $\lim_{n\to\infty} n \int_0^1 x^n f(x) \ dx = f(1)$, where $f(x)$  is continuous, we conclude that 
$$\lim_{n\to\infty}\biggl(n\int_0^{\pi/4}(\tan x)^n\,dx\biggr)=\frac{1}{2}.$$
A: $\newcommand{\bbx}[1]{\,\bbox[8px,border:1px groove navy]{\displaystyle{#1}}\,}
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Laplace Method:

\begin{align}
&\bbox[10px,#ffd]{\lim_{n \to \infty}\bracks{n\int_{0}^{\pi/4}\tan^{n}\pars{x}\,\dd x}} =
\lim_{n \to \infty}\bracks{n\int_{0}^{\pi/4}
\tan^{n}\pars{{\pi \over 4} - x}\,\dd x}
\\[5mm] = &\
\lim_{n \to \infty}\bracks{n\int_{0}^{\pi/4}
\exp\pars{n\ln\pars{\tan\pars{{\pi \over 4} - x}}}\,\dd x}
\\[5mm] = &\
\lim_{n \to \infty}\bracks{n\int_{0}^{\infty}
\exp\pars{-2nx}\,\dd x} = \lim_{n \to \infty}\pars{n\,{1 \over 2n}} = \bbx{\ds{1 \over 2}}
\end{align}
A: You may perform the change of variable $u=\tan x$ to get easily
$$
I_n:=\int_0^{\pi/4}(\tan x)^ndx=\int_0^1\frac{u^n}{1+u^2}du.
$$
Then you may just integrate by parts,
$$
\begin{align}
I_n=\int_0^1\frac{u^n}{1+u^2}du&=\left. \frac{u^{n+1}}{(n+1)}\frac{1}{1+u^2}\right|_0^1+\frac{2}{(n+1)}\int_0^1\frac{u^{n+2}}{(1+u^2)^2}\:du\\\\
&=\color{blue}{\frac12}\frac1{(n+1)}+\frac{2}{(n+1)}\int_0^1\frac{u^{n+2}}{(1+u^2)^2}\:du. \tag1
\end{align}
$$ Observing that
$$
0\leq \int_0^1\frac{u^{n+2}}{(1+u^2)^2}\:du\leq  \int_0^1 u^n\:du=\frac1{n+1}
$$ gives
$$
0\leq \frac{2}{(n+1)}\int_0^1\frac{u^n}{(1+u^2)^2}\:du\leq \frac{2}{(n+1)^2}. \tag2
$$
Then combining $(1)$ and $(2)$ leads to 

$$ \lim_{n \to +\infty}n\int_0^{\pi/4}\!(\tan x)^n\:dx=\lim_{n \to +\infty}n\:I_n=\color{blue}{{\frac12}}$$

as suggested.
