Limits - Blog Posts

best thing you could have ever said to me. like my dad always says. the only way to treat a cold is with contempt. I present: the only way to treat Sad Boi hours is with disdain

[ID: screenshot with the words "And yeah we've all got limits and flaws and whatever"]

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On my studies this week my mentor said I should never write negative things, or things that might bring bad emotions or thoughts to people. Therefore, I deleted a bunch of my posts. Happens that I can be very negative when I am sad or tired. I guess it is human to have negative emotions from time to time, specially when the circumstances you are at, are no the best as possible. Thinking on that and putting that together with many other small conversations I had with people during this time, I decided to change, to go after things I should’ve gone way before and to try to be better at the same time as a person, but not only better to other people, but to myself. I need to be kinder to myself, more comprehensive with my emotions and limits. And I honestly think everybody else should too.

i…. i love it… λ 🥲

is this a valid notation? i wish it were, ill definitely be using it

What’s the worst thing about limits?

• Is it always having to amplify by the conjugate to eliminate indeterminate forms? NO.

• Is it having to learn some limits by heart (especially for trig functions)? NO.

• Is it having to sometimes use L’Hôpital-Bernoulli’s rule over and over again? NO.

• Is it having to just write “lim” before each line? YES. FUCKING YES. WHY OH GOD WHY I HATE THE NOTATION.

The most infuriating part of it is that it’s just a notation and it could so easily be changed, BUT NO, we’re stuck with this shit.

The topologist’s sine curve.

Limts: f(x)=sin(1/x) is a rare example of a function with a non-existent one-sided limit. More technically, f(x)=sin(1/x) is defined for all numbers greater than zero, yet the limit as x approaches zero from the right of f(x)=sin(1/x) does not exist. This can be reasoned by considering the value of f at x-values near zero. Informally, f(near zero) could be 1 f(just a bit closer to zero) could be -1 so f(numbers near zero) does not seem to settle on a single y-value.

Continuity: Note that f is continuous for all numbers greater than zero but not continuous at x=0 since f is undefined there. Even if we were to “fill in the bad point” and let f(0)=0, the function would still not be continuous at zero! (note this is the natural choice as sin(0)=0). We can see that the adjusted f is still not continuous at zero since the sequence x_n=1/(pi/2+npi) converges but f(1/x_n) is the sequence (-1)^n which does not converge. This is similar to the argument above. In other words, closing in on x=0, we can keep finding x values such that f(x)=-1 and f(x)=1.

Topology: In topology, the topologist’s sine curve is a classic example of a space that is connected but not path connected. This space is formed in R^2 by taking the graph of f(x)=sin(1/x) together with its limit points (the line segment on the y-axis [-1,1], the red line on the second image). The graph of f is connected to this line segment as f and the segment cannot be sepearted by an open disc (no matter how small). This can be informally reasoned by the zooming illustration in the second image. But the space is not path connected by the sequence argument above (there is no path to the point (0,0)).

Image credits: http://mathworld.wolfram.com/TopologistsSineCurve.html and https://simomaths.wordpress.com/2013/03/10/topology-locally-connected-and-locally-path-connected-spaces/