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I'm really confused about notions in limit section .

Question 1 : I want to know when we can say that $f(x)$ has a limit ? In other words , what's the necessary and sufficient condition for $f(x)$ that has limit ? Is infinity ($\infty)$ or minus infinity ($-\infty)$ considered as limits ?

Question 2 : What is the a bound ? Why we define bounded or not bounded function when we have limit concept ? What is the application of bound in functions ?

1 Answers 1

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Limit is a local property. In other words, limit of $f(x)$ as $x\rightarrow a$ tells you something about the behavior of $f$ around $a$. While boundedness is a global property. That is, it tells you something about behavior of $f$ over its entire domain.

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    Okay , Can you answer the first question ? – 2017-01-23
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    Typically infinity is considered a limit (or people say f diverges if it has an infinite limit at some point). Whether $f(x)$ has a limit at $x\rightarrow a$ does not depend on $f(x)$ ($f(x)$ need not be even defined). Necessary and sufficient condition for $f(x)$ having a limit as $x\rightarrow a$ is that $\lim_{x\rightarrow a}f(x)$ exists, but that is a tautology. Sufficient condition is continuity of $f$ at $a$ and monotonicity, for example. – 2017-01-23
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    If we have $f(x)$ , how we can determine when $x\to a$ it has a limit or not ? Can you give me a example ? – 2017-01-23
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    @S.H.W There are no fast-and-easy rules I am afraid. The only one is the existence of $\lim_{x\rightarrow a}f(x)$ I mentioned above, but that is a definition. For example, $\sin{1/x}$ as $x\rightarrow 0$ does not exist, $1/x$ as $x\rightarrow 0$ diverges, i.e., has limit $\infty$, while $\sin{x}$ as $x\rightarrow0$ has limit $0$. – 2017-01-23