Limits of the function

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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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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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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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Two types of discontinuity (function limits)

18 January 2012 / Krystian Karczyński

As we all know (for example from my Limits Course), a function is continuous at the point x0 when the left-side limit of this function at this point is equal to the right-side limit of the function at this point and is equal to the value of the function at this point. If any of the equality is not satisfied, the function f(x) is not continuous at the point x0, and the point is called a point of discontinuity. In this naming, you can go a step further and DISTINGUISH the points of discontinuity. See how to do it.

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Proving that sinx does not reach a limit as x approaches infinity

19 September 2010 / Krystian Karczyński

We have a limit of the function as x approaches infinity from the function f(x)=sinx. We intuitively feel that the above limit does not exist. x's are getting bigger and bigger, and the sine values ​​are constantly fluctuating between -1 and 1. But how can we formally demonstrate and prove this?

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Useful limit of a function with cosine

12 September 2010 / Krystian Karczyński

We usually treat the limits of a sine function as: x tends to zero then{sinx} /x=1. Question: what about cosine of x? Does it have any "typical" border?

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Function Limits Calculator (OUTDATED) + New Calculator

10 March 2014 / Krystian Karczyński

Announcement Unfortunately, after over 12 years of providing a free function limits calculator, I had to “turn it off”. The calculator was a simple “widget” from WolframAlpha. Recently, WolframAlpha changed its policy regarding widgets. For example: they no longer calculate “on-site” and instead redirect the user to WolframAlpha. I apologize to all past users of the Function […]

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