That's funny how in France we don't teach L'hospital rule (i learned about it on internet after i got my PhD in maths for example) because we believe it doesn't help students understand what they do, while it's so commonly used as one of the primary calculus tools elsewhere
In Italy they teach it and then strongly advise against using it, and if you use it in a way that it causes circular reasoning (i.e. trying to prove something with L'Hopital that was used to prove L'Hopital itself) they will often not accept it
Once you learn that you can divide by the highest degree term, you also learn that you can just consider that. For example for limit 10 (lim x->inf sqrt(x^2-9)/(2x-6)) you just take sqrt(x^2) on the top and 2x on the bottom, and simplify that to |x|/2x. Now since x approaches positive infinity both |x| and 2x will be positive, so we can just ignore the absolute value and write x/2x = 1/2. That's how they teach it in Italy anyway, IDK about the rest of the world
For 9 pull the to the seventh power outside of the limit which is possible bc the function is continuous. The limit inside is then actually defined as e so it will be e^7 as a result
I like this approach, the only thing to consider is that we can’t say that the limit inside is defined as e. It does end up being equal to e, but that is not how e is defined. It is defined as (1+1/x)^x which is different. To do this method, we would still have to do the limit inside like I did in the video.
Level 10 is way easier if you just consider that x^2-9 approximates x^2 and 2x-6 approximates 2x at large numbers. It simplifies to sqrt(x^2)/2x which is just 1/2
@@dragonizeyta and b doesn't matter if we're taking the limit to infinity/ so it's just limit of |x|/2x and |x| = x if x is positive so it's just limit of x/2x. No need to complicate it like in the video
In my experience, you need to show work like I did in this video. I agree that you can check and see the answer immediately and I even mention that in the video. However, just writing down the answer will often give you no credit, so it is important to go through this question the systematic way so that students can understand