Limits

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Limit of the Function: x to the x to the x … What to Do? (Example with Morals)

07 August 2024 / Krystian Karczyński

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)?

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Limits with Cube Roots (Is it Multiplication by Conjugate?)

05 August 2024 / Krystian Karczyński

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?

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Limits of a Sequence with the Sum of Squares or the Sum of Cubes

05 August 2024 / Krystian Karczyński

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...

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Troubles with Indeterminate Symbols (VIDEO)

24 July 2024 / Krystian Karczyński

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.

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The Symbol 0 in the Limits of Functions and Sequences

27 June 2024 / Krystian Karczyński

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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The Limit of a Sequence with Two Mixed Sums of Arithmetic Sequences

26 June 2024 / Krystian Karczyński

Sequence limits sometimes use formulas for the sum of arithmetic or geometric sequences. It's worse if these sequences are given in a "mixed" way in one line, like here.

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Limit of a Function with Substitution

24 June 2024 / Krystian Karczyński

Harder function limits to calculate often require substitution. Here is an example. Learn detailed methods for calculating function limits using substitution, how to effectively apply them in limit analysis. Discover why some limits do not exist and how to prove it step by step.

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Limit of a Sequence with an Infinite Sum

21 June 2024 / Krystian Karczyński

Let's take the following sequence limit approaching infinity: (1/{12}+1/{23}+1/{3*4}+...+1/{(n-1)*n}). In the problem, we kind of feel that we need to use sequence sum formulas (arithmetic or geometric), but unfortunately, this sequence is neither arithmetic nor geometric... So what to do?

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Limits of Sequences with Logarithms

18 June 2024 / Krystian Karczyński

For the limits of many sequences with logarithms, you can confidently use the transformations and formulas for logarithms learned in high school.

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