one1one1

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.

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.

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.