Not defined * 0 = defined

\[\large \displaystyle\lim_{x\rightarrow 0^{+}} \sin\left(\dfrac{1}{x}\right) = \ \text{Not defined}\]

$\large \displaystyle\lim_{x\rightarrow 0} x\sin\left(\dfrac{1}{x}\right) = 0$

Then how could $[\text{not defined} \times 0 = 0]$ ?

#Calculus #Limits #IITJEE

This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.

When posting on Brilliant:

Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused .
Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting "I don't understand!" doesn't help anyone.
Try to contribute something new to the discussion, whether it is an extension, generalization or other idea related to the challenge.
Stay on topic — we're all here to learn more about math and science, not to hear about your favorite get-rich-quick scheme or current world events.

Markdown	Appears as
`italics` or `_italics_`	italics
`bold` or `__bold__`	bold
- bulleted - list	bulleted list
1. numbered 2. list	numbered list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1 paragraph 2	paragraph 1 paragraph 2
`[example link](https://brilliant.org)`	example link
`> This is a quote`	This is a quote
# I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world"	# I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world"

Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

While the first limit is not defined, it is nevertheless bounded, as it oscillates between $-1$ and $1.$ Thus

$-x \le x\sin\left(\dfrac{1}{x}\right) \le x,$

and so by the Sandwich rule the second limit goes to $0$ as $x \rightarrow 0,$ (from both the left and the right).

Brian Charlesworth - 5 years, 8 months ago

But if we combine both the steps,then they are not following the rules of maths.

Akhil Bansal - 5 years, 8 months ago

We're not really "combining" the steps; we are looking at two different problems. The first limit is undefined since the bounded oscillation of $\sin(\frac{1}{x})$ never ceases or diminishes as $x \rightarrow 0.$ However, in the second limit this oscillation, since it is bounded and multiplied by $x$ , diminishes to $0$ as $x \rightarrow 0,$ i.e., the diminution of $x$ "beats out" the (bounded) oscillation of $\sin(\frac{1}{x}),$ and so the product, and hence the limit, goes to $0.$

Some instances of "undefined times $0$ " will be undefined, but in this case, because of the bounded nature of the "undefined" element, we do in fact find that the limit is defined.

Brian Charlesworth - 5 years, 8 months ago

@Brian Charlesworth – Thanks! I understood now

Akhil Bansal - 5 years, 8 months ago

Shouldn't be surprising. Same holds when you remove the $\sin$ function, IE

$\lim_{x \rightarrow 0 } \frac{1}{x} = \text{ undefined }, \lim_{ x \rightarrow 0 } x \times \frac{1}{x} = 1 \text{ is defined }$

Calvin Lin Staff - 5 years, 8 months ago

Thanks! I got it now.

Akhil Bansal - 5 years, 8 months ago

@Brian Charlesworth , @Calvin Lin , @Prasun Biswas , @Sandeep Bhardwaj ...help please...

Akhil Bansal - 5 years, 8 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$