<s>[INST] Limits connected to a right angled triangle Given a right angled triangle with sides $1,x$ and hypotenuse $y$. Let $\theta$ be the angle contained by side $1$ and hypotenuse. Then evaluate the following limits:

*$\lim_{\theta\to\pi/2}\sqrt{y}-\sqrt{x}$

*$\lim_{\theta\to\pi/2}y-x$

*$\lim_{\theta\to\pi/2}y^2-x^2$ 

*$\lim_{\theta\to\pi/2}y^3-x^3$
I was able to evaluate the third one, using $y=\sec{\theta}$ and $x=\tan{\theta}$.
EDIT:

I also solved the first first and second one. Only second and fourth are is unsolved.

How to evaluate others?
 [/INST] First note that$$y^3-x^3=\dfrac{y^6-x^6}{x^3+y^3}=\dfrac{3y^4-3y^2+1}{x^3+y^3}\ge\dfrac{3y^4-3y^2+1}{2y^3}=1.5y-1.5\dfrac{1}{y}+\dfrac{1}{y^3}$$Also $\cos\theta=\dfrac{1}{y}\to 0^+$ leads to $y\to \infty$ and makes the limit $\infty$ either.
</s>