What is the sine of 1/0? And before you say it's undefined, my calculator doesn't agree with you.?

Question

It'd be written: sin (1/0)

PMP, thanks for the link. The question on my test actually was lim x -->0 for x^2(sin(1/x)). That link actually mentions that a limit exists at that point. I was wrong to leave the x^2 off, I didn't think it would matter.

So a good follow up question is, "Why is the lim as x approaches zero for the function X^2 sin (1/x) = 0?"

Answer 1

It is undefined. Either you keyed it in wrong or your calculator is broken.

Answer 2

Well 1/0 is undefined so I'm not sure what kind of calculator allows you to take the sine of an undefined number. Mine doesn't.

In any case, sin(1/0) is undefined.
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OK you want the limit of x^2 sin(1/x) as x-> 0

The first thing to note is that

lim xsin(1/x) = 1
x->∞

Put another way,

lim sin(x)/x = 1
x->0

You can show this using L'Hopital's rule since

dsin(x)/dx = cos(x) and dx/dx = 1 so rewrite the limit as

lim cos(x)/1 = 1/1 = 1
x->0

The rest should follow.

Visually, this says that as you approach the origin, the slope of y=sin(x) approaches the slope of the line y=x.

Answer 3

As 0<=Ix^2*sin 1/xI<=x^2 as x==>0 x^2*sin (1/x) ==>0

Answer 4

the denominator can't be 0 because it is undefined at that 0

try to sketch the y=1/x graph
the graph will never approach x= o

so for sure sin (1/0) is also undefine

Answer 5

The fact that your calculator produces an answer doesn't necessarily mean that the calculator is correct. 1/0 is undefined. End of story

Answer 6

Well, MATLAB disagrees with your calculator. It says it is not a number. The limit of sin(1/x) as x approaches zero is not defined.

ADDENDUM

The limit of x^2 sin(1/x) as x approaches zero is zero.

Answer 7

Instead of doing math, you are mistakenly relying on a calculator.

Answer 8

My TI-83+ says
ERR: DIVIDE BY 0
1: Quit
2 Goto