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Preface: I'm a calculus student who's a bit frustrated because limits seem like the foundation of calculus and seems full of contradiction. If you could answer my questions & address any misconceptions I have, I would be very grateful. Thank you.

So multiple times I've heard the first thing you do want to do to derive the limit of a function is to plug it in.

This drives me crazy since the definition of a limit is what happens around the point not at it but I understand this works for continuous functions e.g. algebraic & transcendental functions are continuous everywhere they are defined.

However sometimes when we plug in values to the function we get an indeterminate form of "0/0" or "inf./inf." or any of the other 6 indeterminate forms. What does that exactly mean? Does this mean that the limit definitely exists & we could find it by algebraic manipulation?

I know that when we find limits and we get the undefined form of "constant/0" that the limit doesn't exist, but are there any other undefined forms in calculus when we know the limit doesn't exist?

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In the context of limits, we call expressions such as $\infty \cdot \infty$ $\infty \cdot 0$, $\infty/\infty$, $0/0$, $\infty/0$, $\infty + \infty$ and $\infty-\infty$ indeterminate forms. We can potentially use L'Hôpital's rule to evaluate limits involving $\infty/\infty$ and $0/0$, but there are often other tools available to us depending on the limit in question.

I'm not sure what you mean by "undefined form," just know that any expression involving $\infty$ or division by zero is not a real number. An indeterminate form indicates that you'll either have to do some extra work beyond substitution to evaluate the limit, or the limit might not exist at all. Proving that a limit does not exist is often easier than computing a limit directly due to the definition of convergence.

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