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If $\ f:\mathbb R\rightarrow \mathbb R\ $ where $f(x)=\frac{1}{x^2}$ then $$\lim_{x\to0}f(x)=?$$
Which of the following option is most correct among these
(a)$\infty$
(b) limit does not exists
Solution
According to me$$\lim_{x\to0^+}f(x)=\lim_{x\to0^-}f(x)=\infty$$
but the limit is not finite so we can say that the limit does not exists
I am confused to choose the option among (a) & (b) in single choice type question
Can anyone tell me ?
The limit in fact does not exist, as $\infty$ is not a value but more a kind of expression. But I think everyone will know that by saying $\lim_{x\to 0}f(x)=\infty$ you mean $f(x) \to \infty$ for $x\to 0$.
When beginning to work with limits it is didactically-wise more meaningful to use the the latter, but later on it doesn't matter.
The limit does not exist in the sense that there is no real number $L$ for which the formal definition of $f(x) \to L$ as $x \to a$ (with $a=0$ in your case) is satisfied; you have probably seen this $\varepsilon \; \delta$-definition and if a function satisfies it, the following notation is used: $$\lim_{x \to a} f(x) = L$$
Usually however, the concept of limits is generalized to capture some other kinds of (limit) behavior. One of those is $f(x) \to +\infty$ as $x \to a$. Note that this has a different formal definition. Your example satisfies this definition, so if you have seen this generalization: it will be the answer and the following notation is introduced: $$\lim_{x \to a} f(x) = +\infty$$
The limit is correct, the definition says that a limit of a function $f$ does not exists if and only if doesn't exists the limite finite in $\mathbb R$ and the limit $\pm\infty$. So $lim_{x\to x_0 }f=l$ exists if $l \in \mathbb R, $ or $l= \pm\infty$. Hope this help you.
Technically, (b) is the most correct. However, we've invented a shorthand for the times when the function grows larger and larger without coming back down as $x \to 0$. That shorthand is written $\lim_{x \to 0}f(x) = \infty$. Keep in mind, though, that it is a shorthand, and it means that for any $N>0$, we may find a $\delta > 0$ such that $|f(x)| > N$ for all $0<|x|<\delta$. That is a fundamentally different statement from $\lim_{x \to 0}f(x) = L$ for any finite $L$.
Usually we say that a limit exists if it is equal to a number. This is the convention used in, for example, Stewart's Calculus book. Infinity ($\infty$) is not a number and so if a limit is equal to $\infty$, then it does not exist. With this convention the answer is (a) and (b).
Now, this is the standard convention, but there are probably books/notes/teachers who use other conventions. So to accurately answer your question, you need to ask your teacher or consult your textbook/notes.
