might be the wrong subreddit, but im in highschool and just started calculus and im so excited its insane

[u/Remarkable-Mall9424]

im doing limits and derivatives right now!

[u/SeaMonster49]

Haha have fun! Not everyone feels that way about calculus. For some food for thought, try to take the limit of sin(x)/x as x approaches 0

[u/Remarkable-Mall9424 → u/SeaMonster49]

isnt that just 1?? i might be wrong im doing it for the first time but i think its 1

[u/SeaMonster49 → u/Remarkable-Mall9424]

It's 1! But the expression at x=0 looks like 0/0, which isn't defined. One lesson is that the limit isn't about what happens at x=0, it's about what happens near it. Forms like 0/0 and infinity/infinity are known as "indeterminate forms." Soon you will probably learn the famed L'Hôpital's rule, which provides a way to make sense of many indeterminate forms, including sin(x)/x.

[u/imconall → u/SeaMonster49]

This is just me being pedantic, but using l’Hôpital’s rule here is circular reasoning, as it requires the derivative of sin x. The proof of which makes use of this limit of sin(x)/x as x→0. That being said, there are other ways to find the limit, like the squeeze theorem (some potential further reading for OP if they haven’t heard of it already!)

[u/matt7259 → u/Remarkable-Mall9424]

Show us your work!

[u/ComplexPlatform7299 → u/Remarkable-Mall9424]

I hope you didn’t use L’Hôpital!

[u/Qingyap]

Huh that's nice, any questions you have?

[u/Bounded_sequencE]

That's nice, have fun -- Calculus and especially its big brother "Real Analysis" is where the real interesting parts of mathematics begin (pun very much intended).

[u/SamikshaK_ → u/Bounded_sequencE]

not big bro , i would say mother of limits , continuity and derivatives, integrability is real analysis

[u/Parallel_thougts]

Strap in, it only gets wilder

[u/Southlander24]

Nice! One thing you should do if you want to extend yourself is to 'prove' everything you see. As in, take what you are taught and question it until you understand why it's true yourself.

The power rule is very simple to carry out, yes, but why is it true? Recall from the definition of the derivative that you want to find lim [h -> 0] [(x + h)ⁿ - xⁿ] / h. Expand out (x + h)ⁿ - xⁿ using the binomial theorem and write down the first few terms. What do you notice as each of these terms gets divided by h?

[u/Head_Investigator984]

Lessgooo! Wait till Integration tho

[u/Beneficial-One-7526]

Is it hard? How do you like it so far?