As a curiosity, I’ll calculate the limit of the function: (x^x^x – cosx).
More interesting than the calculations themselves are the two morals that can be drawn from them. But the morals are at the end (has anyone ever seen a story with a moral at the beginning)?
When given a limit to calculate that involves some subtraction with a root (and which obviously couldn’t be calculated more simply), such as: “Something – square root of something”, “square root of something – Something”, or “square root of something – square root of something”, we used a trick I call – “multiplication by the conjugate.”
We simply multiplied this expression by its counterpart with a plus sign, or rather by a fraction where this counterpart was in the numerator and the denominator.
What to do when the roots are cube roots?
In the limits of sequences, sometimes the expression contains the sum of squares or cubes of consecutive natural numbers. What then?
The answer is simple: formulas for the sum of squares and the sum of cubes of consecutive natural numbers. They go like this…
This post is a kind of response to the question:
“I don’t understand something and I need an explanation, why do you simplify ‘n’? I mean that n/n is an indeterminate symbol (infinity over infinity) help because I’m already lost with this.”
Understanding what indeterminate symbols REALLY are can be quite tricky. It also raises many questions about what you “can” and “can’t” do with them.
When working with limits of functions and sequences, we frequently come across various mathematical oddities, such as: 0/0 or 1/0.
The question is – what do they mean? I often come across completely incorrect opinions like: “at higher levels in mathematics, you can divide by 0”.
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