Quick question. For F(X)= X/Sin(X), is there a hole at X=0?
What do you get when plugging 0 into F(X) ?
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
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?
Thanks so much! Can you help me explain why there is an oblique asymptote?
wot's an oblique asymptote?
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
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.
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.
Ok, so NO oblique asymptote, correct?
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 limx -> 0 x/sinx ?
A slant asymptote
So that's only at infinity?
and also negative infinity if the domain goes there too.
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.
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