when one uses the approach of nonstandard calculus. However, these shortcuts can be easily adopted in a standard calculus course.
Let me elaborate.
\MLXC Derivatives are similarly computed without taking a limit. Continuity of functions is defined as \ows to infinity is Represented by an "unbounded hyperreal number", the sequence Could Quickly Grow, or very Quickly, or slowly, or very slowly, and for Each Conceivable "growth rate" there is a hyperreal number of the Corresponding "order of magnitude ". There is an uncountable number of Different Growth rates. The set of hyperreals Complicated is quite a thing to imagine! No wonder a textbook Can not Give specific examples of infinitesimals and Keeps writing "dx" or "epsilon." However, students will Have a hard time Trying to Understand this concept Without an intuitive picture. I think it is Not Productive to require students to Understand the hierarchy of Growth Such orders at an early stage in a beginner's calculus course. I see this as the first major problem. The secon d problem is that not all limits can be computed simply by \third problem is That I Will Have students trouble with the notion of convergence of series after seeing all WHERE Calculations So Many Powers of "epsilon" are Automatically infinitesimal. The convergence of series is an Especially shady spot in nonstandard analysis. That says a truncated one series has the sum S (n) Where n is the number of terms. Then one Substitutes an "unbounded integer" instead of n. This is impossible to do in practice unless Already one has S (n) as an analytic function of n. Also, the idea of "unbounded integers" is very complicated. For example, "cos (2 * Pi * n)" should be "almost equal" to 1 When n is an "unbounded integer." (Or Should it be
Exactly equal to 1?) However, as we Have Seen, "cos x" is undefined Actually if x is unbounded, like when to x = 1/epsilon with infinitesimal epsilon. It appears that \e bounded function, then x=y when we finally take the limit epsilon=0.\
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