one1one1 · HackerTrans

If your daughter is not going to learn that 1) x^2 is much smaller than sqrt(x) for x small. Or to know how 2) how to computer the tangent line of y=sin(2x), then I think she will not be able to appreciate how to approximate the solution for a problem and bound the error of that solution, and that are very useful skills. If the learns those two facts I think she will be able to solve that limit and appreciate how rules compose to apply your knowledge more broadly.

one1one1

4 anni fadiscuss

Perhaps you don't need to solve any limit because computers can do that for you, but when you try to solve a problem in your mind if you know limits you don't need a computer all the time, you just need to know that some approximations give you an acceptable solution to certain problems and in many cases the approximation is based on a limit. For example in many problems one can foresee the capacity of a process and what kind of error are acceptable is that setting, so you are able to obtain local solutions or assymptotic expansions that save you job.

one1one1

4 anni fadiscuss

You don't need tricks to solve the limit of (x^2)sin(2x)/sqrt(x) as x->0, only to know equivalent infinitesimals, so that sin(2x) can be replaced by 2x in this limit and 2x^3/sqrt(x) can be simplified to 2x^(3-1/2) so the limit is zero. Moreover, you should know that x^2 and sin(2x) are really smaller compared to sqrt(x) so that the limit is zero. Knowing how to compare two infinitesimals can help you to replace a difficult process for an easier one, just like in differential equations sin(x) is replaced by x for small vibrations. That kind of knowledge allows you obtain an approximate solution in many real processes. So I don't consider that knowledge a bag of tricks. Furthermore, replacing sin(2x) for 2x is just the process of taking the linear approximation (the tangent line near x=0 for y=sin(2x)), so this is just an example of the power of calculus to replace a difficult question with an easier one. Also you learn to explore the world near a point (like a miscroscope) by the process of taking the linear approximation or taylor expansion of a function near a point, then you iterate this procedure to nearby points just like what happens when you solve a differential equation by Euler method. So I should say that there is a framework to solve problems via approximate solutions and computing limits is just a good first step to enter the realm of Calculus.

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