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I know of these ones:

1) left side limit doesn't equal right side limit

2) value of x is in limit approaching the end of closed interval

3) the function is oscillating infinitely when approaching the limit

4) the limit isn't approaching any particular value (it's "infinite")

5) by the definition (but it's a tedious and slow way of checking the existence of a limit, so I'm looking for faster methods if there are any)

I have troubles applying these rules/methods to actually find if a limit is non-existent. For example: $$\lim_{x\to 0}\frac{\sqrt{x+1}-1}{x^2}$$ and $$\lim_{x\to 0}\frac{(1-\cos x)^2}{\tan^3 x-\sin^3 x}$$ Which method to use (of the above ones or if there are any other ones) to quickly check and see that these 2 limits actually don't exist?

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    What do you mean by condition ``2)``? – 2017-01-22
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    I'm sorry, english is not my first language, but what I meant was for example $\lim_{x\to 0}\sqrt{x}$ doesn't exist because the square root is defined for only non-negative numbers. So 0 is the end of the interval where the function is defined. Basically left side limit of this function does not exist thus the limit doesn't aswell. – 2017-01-22
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    Different textbooks may have different opinions in that situation, but many would say (and I agree) that $\lim_{x\to 0}\sqrt{x}$ *does* exist. See discussion here, for example: http://math.stackexchange.com/questions/1544273/limit-definition-when-right-hand-limit-has-nonempty-domain-but-left-hand-limit-h – 2017-01-22

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