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If Right-hand derivative(RHD) and left-hand derivative(LHD), both $\to \infty$ then can we say derivative exists. As they are limits, if both are equal we should say that derivative exists. But I am getting different interpretations at different sources. So what is the standard definition?

Praveenkumar Donga
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3 Answers3

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If the right-hand side derivative diverges to infinity, it does not exist. Hence, the derivative also does not exist. This simply follows from the definition of the limit of a function and the definition of "divergence to infinity".

Remember that for some function $f:\mathbb{R}\to\mathbb{R}$, $$\lim_{x\to\infty} f(x)=\infty$$ is just an abuse of notation to state that the function grows without a bound or more precisely:

For all sequences $(x_n)_n$ in $\mathbb{R}$ that diverge to infinity, also the sequence $(f(x_n))_n$ diverges to infinity.

In case you have forgotten, we say a real sequence $(x_n)_n$ diverges to infinity, if for any $M>0$, there exists an $N \in \mathbb{N}$ such that for all $n \geq N$, we have $x_n > M$.

In particular, $\lim_{x\to\infty} f(x)=\infty$ does not mean that the limit exists.

Toni
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  • I got your point and I just want to make sure that, according to your reply are the notations limit divergence to $\infty$ and limit doesn't exist same? In literature, they both are being used separately. – Praveenkumar Donga Apr 11 '20 at 12:13
  • No, one is just a special case of the other. There are other reasons why a limit might not exists. For example, for $\sin$ the limit as $x\to\infty$ does not exist, but it is also bounded from above by 1. – Toni Apr 11 '20 at 12:17
  • So limit doesn't exist cases: i) when $\lim_{x \to a+} \ne \lim_{x \to a-}$ ii) when one or both LHR and RHL are undetermined or $\infty$ iii) ${f(x_{n})}{n}$ doesn't converge to f(a) when ${x{n}}$ converge to $a$. Basically when $(\epsilon \delta)$ definition of limit was not satisfied we would say limit doesn't exist. – Praveenkumar Donga Apr 11 '20 at 12:30
  • Yes, the last sentence is correct. Note that the epsilon-delta definition is equivalent to the definition involving sequences that I gave in my answer. So, if one of them (and hence both) are not satisfied we say the limit does not exits. Special cases are: (i) the RHS limit and LHS limit exist, but are unequal (ii) divergence to $+\infty$ or $-\infty$. – Toni Apr 11 '20 at 12:34
  • Two more equivalent definitions I have found for limit i) Cauchy criterion : $ \forall \varepsilon \gt 0 \exists$ nbd (N) s.t $\mid f(x_{1} -f(x_{2})\mid \lt \varepsilon \forall x_{1}, x_{2} \epsilon N; x_{1},x_{2} \ne a$) ii) If Four Functional limits at a point are equal, their common value is the limit of f at that point. – Praveenkumar Donga Apr 11 '20 at 12:48
  • If the right-hand side derivative diverges to infinity, it does not exist. --- I think you mean "difference quotient" and not "derivative". There is a difference between difference quotients approaching an infinite value (and YES, the notion of an infinite derivative existing is certainly meaningful -- see this question and this google search) and the derivative function itself approaching an infinite value (and your abuse of notation comment isn't really valid). – Dave L. Renfro Jul 31 '22 at 05:29
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we say that f is differentiable at c if the limit $\lim_{x\to c} \frac{f(x)-f(c)}{x-c}$ exists. By existence of limit we mean a limit equals a finite real number (in case of real valued functions on real variables). RHD & LHD both tend to infinity means the one sided limits fails to exist and thus we can conculde that above defined limit does not exist or simply f is not differentiable.

Ravindra
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The very basic defination about limit is that if a sequence converges then it's limit is unique . But when a function limit is infinity we say that the function "diverges" to infinity . So it isn't unique so we can't say that limit exist .

Shaswat Pratiyush
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