$ \lim_{a\to\infty} {a\int_{0}^{\frac{\pi}{4}}e^x\space \tan^a{x}\space dx}$ $$Let \space I(a) = \int_{0}^{\frac{\pi}{4}} {e^{x} \tan^{a} x} \space . \space Find 
\space \lim_{a\to\infty}aI(a). $$
Well this is an integral that I'm curently dealing with and so far I've tried to solve it using different approaches. Still, I haven't solved it and what I find always is an undefined answer. Even when I used mathematica it returned an undefined answer. This solution in my opinion is most likely to be the one that works: $$\\$$ Our Integral is $$\int_{0}^{\frac{\pi}{4}}e^x\space \lim_{a\to\infty}[{a\tan^a{x}}]\space dx,$$
letting $$A=\lim_{a\to\infty}{a\tan^a{x}}$$
We know that $$x \in [0,\frac{\pi}{4}]\space \Rightarrow \space \tan{x} \in[0,1] \space \Rightarrow \space \lim_{a\to\infty}a\tan{^ax}=
\  \begin{cases} 
      0 & 0\leq x< \frac{\pi}{4} \\
      1 & x=\frac{\pi}{4}\\
   \end{cases}
  \
$$
Which tells us that
$$ A =
\  \begin{cases} 
      0 \times \infty & 0\leq x< \frac{\pi}{4} \\
      \infty & x=\frac{\pi}{4}\\
   \end{cases}
$$
Which are some undefined integrands... . Mathematica says the integral is undefined but the book, Advanced Calculus Explored, says it has an answer.
I think this is the right solution and I just need to work on the limit a little bit more, but this is something that I'm stuck in. I appreciate any kind of hints or helps.

Rewrite the integral as
$$\int_0^{\pi/4} dx \, \exp{x} \, \exp{\left [a \left (\log{\tan{x}} \right ) \right ]}$$
Sub $x=\pi/4-y$ and use the tangent addition rule and the integral is
$$\exp{\left ( \frac{\pi}{4} \right )} \int_0^{\pi/4} dy \, \exp{(-y)} \, \exp{\left [a \left (\log{\left (\frac{1-\tan{y}}{1+\tan{y}} \right )} \right ) \right ]}$$
Not that the dominant contribution to the integral as $a \to \infty$ is in the interval in a small neighborhood about $y=0$.  So expand the integrand about $y=0$; then the interval of integration may be expanded out to $[0,\infty)$ because the other contributions are exponentially subdominant.  Accordingly, as $a \to \infty$, the integral behaves as
$$\exp{\left ( \frac{\pi}{4} \right )} \int_0^{\infty} dy \, e^{-2 a y} = \frac1{2 a} \exp{\left ( \frac{\pi}{4} \right )}$$
Accordingly, the sought-after limit is $$ \frac12 \exp{\left ( \frac{\pi}{4} \right )}$$
This has been verified in Mathematica.
