Borbin the 🐱

The final one:

To evaluate the limit, we apply standard trigonometric identities to decompose the expression into terms involving sin⁡(x)/x​ for which a limit for x→0 exists.
While algebraic manipulation to eliminate the denominator is sometimes effective, it is not applicable in this case.
Instead, recognizing and leveraging this fundamental limit allows for a straightforward and elegant solution.

Here is the third one:

Option c) leads to the solution. The expression
f(x) + f(y) + x²y + xy²
is part of the expansion of the binomial
(x+y)³
If f(x) corresponds to the first or last term of the binomial expansion, the equation can be solved.

Here is the second one:

Start by moving the x term to the right-hand side and setting x=0. This is the first equation for b.
Then with b in place, get the second equation. Isolate a and substitute 0 for x. This gets

I've been helping out with high school math lately, and some of the questions are quite interesting.
Check this one out:

The key is to derive two equations from the derivative. With a fourth-degree polynomial, we get three points where the slope is the same, but only two of them share a common tangent line.
After that, everything falls into place.

August 21st 2017
Sun Eclipse over North America

Using the Optrel Weldcap to watch the sun eclipse seemed like a great idea.

Worked great to begin with, but unfortunately the Weldcap failed after 10min and the auto darkening switched on and off.
Weldcrap! This was quite a unsafe experience. Best thing to do was this: