Yeah, that works. The only reason I did so otherwise is because generally these limits are introduced at the beginning of an introductory calculus course, before students have had derivatives or L'Hopital's Rule. At least, that is how it was for me.
It is usually done without calculus. Just because it is used to define the derivatives of certain functions. If you compare the areas in the graph you can come up with the inequalities and use the squeeze theorem. Probably works out to the same formula above, but with a little more motivation.
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