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In the first few lessons of Calculus I, we are told to solve limits algebraically and do not use l’Hôpital's Rule. Although, if the teacher does not specify which method to use, or not to use, then yes, use l’Hôpital's Rule because it is faster!

This is exactly what I have to do for teaching calculus! It's neat that we use limits to evaluate derivatives and then use derivatives, in the form of l'Hôpital's Rule, to solve limits.

That is the correct answer!

Yes, use l’Hôpital's Rule if the question does not tell you not to use it! In the calculus courses I took and have to teach, we could only use algebraic techniques and were not allowed to use l’Hôpital's Rule in the unit about limits.

I think actually the conjugate is way faster because of the x - 7 cancellation but I like how l'hopital is more general so honestly me too

Neat! Multiplying by the conjugate may be faster because the root differentiates by the Chain Rule, which brings in more factors to deal with and makes l’Hôpital's Rule a bit complicated. Alternatively, we can try a mix of both methods! Start with multiplying by the conjugate, then as long as the indeterminate form is appropriate (±∞/±∞ or 0/0), use l’Hôpital's Rule. Whatever method you prefer, use the method that works fastest and most accurately for you and is also allowed by the exam instructions.

Good question, I actually don't know! I think factoring to solve limits is typical of Advanced Placement (AP) calculus and definitely university/college calculus. According to page 40 of “AP Calculus Course and Exam Description” (apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf), “[f]actoring and dividing common factors of rational functions” looks to me like this video is typical of what is on the AP curriculum.

There is a lot of annoying algebra! For some reason, I had to learn this as a student and is also in the curriculum I teach in multiple universities. L'Hôpital's Rule is easier and faster.

Good question! When we multiply (x − 7) and (x + 7), we have the difference of squares x² − 49, which does not cancel much. Instead, if we target the awkward square root, then we can eliminate the square root as a difference of squares.

Good call! It is! I think this is why calculus courses begin with limits because then we can extend limits to the limit definition of the derivative.

I am not officially a professor because I am unemployed. I do things that professors do, like teaching, supervising undergraduate/graduate research projects, and evaluating funding applications. Thank you for watching!

Absolutely, l’Hôpital’s Rule is another way to solve this limit question! In fact, l’Hôpital’s Rule is applicable for limits of the form “0/0” (what we have here) and “±∞/±∞”. In the beginning lessons and midterm(s) in Calculus I, instructors tend to tell students to solve limits using algebraic methods rather than l’Hôpital’s Rule. However, if the question does not specify which method to use, l’Hôpital’s Rule can be an easier and faster method than algebra.

That is great to be able to solve this question mentally (by inspection)! Solving mentally is especially suitable on questions where it is not necessary to show calculation work, like multiple choice questions.

You are right! L'Hôpital's Rule is perfect for indeterminate forms “0/0” (the situation here) and “±∞/±∞”. In the calculus courses I took and taught, we had to solve limits algebraically. On my midterm exam, the instruction specifically told us to not use L'Hôpital's Rule. It was not until later on in the course, about applications of derivatives, that we had the option of L'Hôpital's Rule.

Thank you. Sorry for the advance@@DrYheartLab

I am glad you brought up l’Hôpital’s Rule! If the question does not specify which technique to use, or not to use, l’Hôpital’s Rule is often a faster method than algebra.

Amazing, @Jan-W, I could not have thought of your solution! Absolutely, write the “x − 7” as “(x + 2) − 9” and then factor as a difference of squares to cancel with the numerator. I am glad you brought up this alternative solution! 💡

Thats splendid. I would not be able to come up with that in 1000 years!!

Not to worry! There may be multiple ways to solve limits. Just use what works well and accurately for you.

It can be quite an abstract topic! The usual example is the sodium-potassium pump. I included some uncommonly used examples to show that there are other ways to use ATP and also sources of energy other than ATP, like the ATP-binding cassette transporter, light-gated ion channel, and electron transport chain. Let me know if there is another method of presenting the subject that is more understandable!

Glad you asked! Arrhenius acids are a subset of Brønsted-Lowry acids, which are a subset of Lewis acids, the most general category of acids in some university programs. All the acids in this video are Arrhenius acids and thus are also Brønsted-Lowry acids. The Arrhenius definition of acid is quite narrow about 1) hydrogen ion (H⁺, proton) and 2) in water. The idea of a free H⁺ floating in solution is not the best description because a free H⁺ is too unstable. The Brønsted-Lowry definition is broader; acids are H⁺ donors in water and also in other environments, such as the gas phase. Also, the Arrhenius definition does not say where the H⁺ goes. Brønsted-Lowry-type of reactions necessarily involve an acid-base pair of reactants. The H⁺ does not float around like in the video for the Arrhenius definition. According to Brønsted-Lowry, the HCl in water has to donate H⁺ to something. If there is no other H⁺ acceptor other than water, then water is the reactant that accepts H⁺ from HCl. The free H⁺ is too unstable, but H₃O⁺, a water molecule that accepted H⁺, has been observed.