Is There a Hole at X=0 for F(X)= X/Sin(X)?

[CanadianEh]

Hi there,
Quick question. For F(X)= X/Sin(X), is there a hole at X=0?

Thanks.

 

[Bohrok]

What do you get when plugging 0 into F(X) ?

 

[tiny-tim]

Hi there,
Quick question. For F(X)= X/Sin(X), is there a hole at X=0?

Thanks.

Hi CanadianEh!

At x = 0, obviously, it's 0/0, which is undefined (it's known as an "indeterminate form"), so yes in that sense there's a hole …

of course, F(x) does tend to a limit at as x -> 0

 

[CanadianEh]

0/Sin 0 = undefined.

So basically, there's my answer. There is a hole at x=0. There is also an oblique asymptote of f(x)=x, correct?

 

[CanadianEh]

Hi CanadianEh!

At x = 0, obviously, it's 0/0, which is undefined (it's known as an "indeterminate form"), so yes in that sense there's a hole …

of course, F(x) does tend to a limit at as x -> 0

Thanks so much! Can you help me explain why there is an oblique asymptote?

 

[tiny-tim]

Thanks so much! Can you help me explain why there is an oblique asymptote?

uhh?

wot's an oblique asymptote? ​

 

[CanadianEh]

When a linear asymptote is not parallel to the x- or y-axis, it is called either an oblique asymptote or equivalently a slant asymptote.

In the graph of X/Sin(X), there appears to be an asymptote at y=x

 

[chroot]

The function continues to have a defined value as you get arbitrarily close to zero, thus the limit as x->0 is defined. The function itself is undefined only exactly at zero.

- Warren

 

[Bohrok]

Try graphing x/sin(x) and you'll only see vertical asymptotes when the denominator, or sin(x), is 0.
As far as I know, a rational function P(x)/Q(x) where P and Q are polynomials has an oblique asymptote only when the degree of the numerator is one larger than that of the denominator. In x/sin(x) you have a transcendental function in the denominator.

 

[CanadianEh]

Ok, so NO oblique asymptote, correct?

 

[tiny-tim]

When a linear asymptote is not parallel to the x- or y-axis, it is called either an oblique asymptote or equivalently a slant asymptote.

In the graph of X/Sin(X), there appears to be an asymptote at y=x

Still totally confused as to why this is called an asymptote instead of a tangent.

Anyway I can't see how it's slanting ……

what is lim_{x -> 0} x/sinx ?

 

[Bohrok]

Ok, so NO oblique asymptote, correct?

That's right.

uhh?

wot's an oblique asymptote? ​

A slant asymptote
http://home.att.net/~srschmitt/precalc/precalc-fig12-03.gif

 

[tiny-tim]

A slant asymptote

So that's only at infinity?

 

[Bohrok]

and also negative infinity if the domain goes there too.

 

[HallsofIvy]

tiny-tim, the word "asymptote" was wrong here. He intended "tangent" as you suggested. Because there is a "hole" at x= 0, there is no tangent there.