Determine the limit of each function analytically. Show neat and complete soluti

Find the integral of ${\displaystyle \sec \left(x\right)}$

Consider ${\displaystyle f\left(x\right)=x{e}^{-kx}}$ for ${\displaystyle k>0}$. Find the relative extrema and the points of inflection of the function.

Calculus question about the existence of antiderivative
Let f(x) be the function equal 0 on ${\mathbb{R}}_{<0}$ and equal 1 otherwise. Is clear to me that f has derivative undefined at $x=0$. Is intuitive because the derivative would be $\mathrm{\infty }$ as f increases by 1 in an infinitely small space.
Now I was wondering whether antiderivative exists at 0. My thoughts: antiderivative corresponds to area under the function. Before zero the are is always zero and after zero the area is 1 so that like in derivative case the area increases by 1 in an infinitely small space. But I don't know if intuition can be used in this case.
When does antiderivatives of functions exists in general and when not?

Evaluate the following limits. $\underset{(x,y)\to (1,-2)}{lim}\frac{y^2+2xy}{y+2x}$

Evaluate the following limits.
$\underset{(x,y)\to (-3,3)}{lim}(4x^2-y^2)$

What is the equation of the normal line of ${\displaystyle f\left(x\right)=-x^4+4x^3-x^2+5x-6}$ at x=2?