Determine the limit $\lim_{k \to \infty} \int_{0}^1 x^{- \frac{1}{2}}\cos(x^k)e^{-\frac{x^2}{k}} dx.$ 
Determine the limit $$\lim_{k \to \infty} \int_{0}^1 x^{- \frac{1}{2}}\cos(x^k)e^{-\frac{x^2}{k}} dx.$$

I suppose the dominated convergence theorem would be in place here? If I denote $f_k(x)= x^{- \frac{1}{2}}\cos(x^k)e^{-\frac{x^2}{k}}$, then $$f_k(x) = x^{- \frac{1}{2}}\cos(x^k)e^{-\frac{x^2}{k}} \le x^{- \frac{1}{2}}e^{-\frac{x^2}{k}} \le e^{-\frac{1}{k}}$$
So now I would have that $$\lim_{k \to \infty} \int_{0}^1 x^{- \frac{1}{2}}\cos(x^k)e^{-\frac{x^2}{k}}dx = \int_0^1 e^{-\frac{1}{k}}  dx = e^{-\frac{1}{k}} ?$$
Am I missing something here or is it really $e^{-\frac{1}{k}} $?
 A: Mistakes in attempt

*

*$x^{-\frac 12}e^{-\frac{x^2}{k}} \leq e^{-\frac 1k}$ is false, for example as $x$ approaches $0$ the LHS approaches infinity while the RHS stays fixed.


*The limit is incorrect on the last line.

To apply the DCT , we want to make sure that we know whether the function $\lim_{k \to \infty} x^{-\frac 12}\cos(x^k)e^{-x^2/k}$ exists or not as an a.e. pointwise limit on $[0,1]$, and what it is.
To do this, we look at the parts.

*

*$x^{-\frac 12}$ doesn't depend on $k$, so it stays as is.


*What does $x^k$ look like for large $k$ and $x\in [0,1]$? What is this part going to converge a.e. to?

 It would be the function $f(x) = 0$ for $x \neq 1$, and $f(1) =1$. So it's a.e. zero, in other words.


*

*Hence, what does $\cos(x^k)$ converge a.e. to?


 Since the cosine is continuous, $\cos(x^k)$ would converge to $1$ a.e.


*

*Where does $-\frac{x^2}{k}$ converge to a.e. ?


 Obviously it converges to the zero function.


*

*Hence, where does $e^{-\frac{x^2}{k}}$ converge?


 Since the exponential is continuous, close to $1$.

Thus, putting things together, we get by the product rule that $\lim_{k \to \infty} x^{-\frac 12}\cos(x^k)e^{-x^2/k}$ equals:

 $x^{-\frac 12} \times 1 \times 1 = x^{-\frac 12}$ a.e.

and thus, providing the DCT held, the answer is:

 $\int_0^1 x^{-\frac 12}dx = 2$.


How to think about the DCT? We obviously want to bound the function $|x^{-\frac 12}\cos(x^k)e^{-x^2/k}|$ uniformly in $k$ and get an integrable function of $x$ as the bound.

*

*What is the obvious bound for $|\cos(x^k)|$?


 It's $1$, isn't it?


*

*What is an upper bound for $-\frac{x^2}{k}$ for any $k$ and $x \in [0,1]$?


 Come on, they're all negative quantities, so $0$!


*

*Thus, what is an upper bound for $e^{-\frac{x^2}{k}}$?


 By monotonicity of the exponential, the required upper bound is $e^{0}=1$.


*

*Leaving the first term as is, what does the eventual bound look like? Is it integrable?


 $x^{-\frac 12} \times 1 \times 1 = x^{-\frac 12}$, obviously integrable!

Of course it turns out to be integrable, but it also turns out to be ... yeah, it's in the spoiler.
Here's the integral for $k=25$ computed via Wolfram Alpha, and you can check that it's already quite close to the actual answer we derived.
A: On the interval $(0,1)$ the integrands, which are positive, converge pointwise to $1/\sqrt x.$ Note that $1/\sqrt x\in L^1(0,1).$ Furthermore, each of these integrands is bounded above by $1/\sqrt x.$ We're done by the DCT: The limit is $\int_0^1 1/\sqrt x\, dx = 2.$
