Indeterminate forms (1 Viewer)

[leehuan]

What makes them indeterminate anyway. What's the problem saying
1^infinity=1
0*infinity=0
0^0=1
infinity^0=1

Only some of them make sense to me, such as infinity-infinity

 

[Paradoxica]

Because there could be a term that approaches zero, but there is also a power that magnifies the number more than it shrinks.
Take the limit definition of e as an example.

Evaluating the limits naively results in 1, however, plugging large n into your calculator shows a number that approaches 2.718281828459045......

 

[Paradoxica]

Furthermore, for product limits, one of the product functions might asymptotically blow up exactly as fast as the other term converges to zero.
For example
, even though one factor goes to infinity and the other goes to zero.

 

[Paradoxica]

Similar reasoning exists for your other two categorical limits. I have not the virtue to demonstrate the fallacious reasoning of those limits you created, but I'm sure easy counter examples exist.

 

[InteGrand]

What makes them indeterminate anyway. What's the problem saying
1^infinity=1
0*infinity=0
0^0=1
infinity^0=1

Only some of them make sense to me, such as infinity-infinity

Here is the definition of indeterminate form (from https://en.wikipedia.org/wiki/Indeterminate_form ):

"In calculus and other branches of mathematical analysis, limits involving an algebraic combination of functions in an independent variable may often be evaluated by replacing these functions by their limits; if the expression obtained after this substitution does not give enough information to determine the original limit, it is said to take on an indeterminate form."

So to show the above are indeterminate forms, you can come up with an example of two different limits where direct substitution results in those indeterminate forms, and the two limits are different.

 

[InteGrand]