[maths] Nelson's Reply to Ned's post
- From: "P. R. Stanley" <prstanley@xxxxxxxxxxxx>
- To: maths@xxxxxxxxxxxxx
- Date: Sat, 04 Nov 2006 14:06:34 +0000
Ned,
No, sec A does not approach tan A as A approaches 90 degrees; both
of =
them grow without bound as A approaches pi/2. However, both curves
are =
asymptotic to the vertical line at A =3D 90 degrees (as well as to each
=
other); i.e., the distances between them approach 0.
1/x approaches 0 (in the limit) as x grows infinite; 1/x
approaches 2 =
in the limit as x approaches 1/2, and indeed, it equals 2 when
x=3D1/2.=20
Don't confuse limits with asymptotes although two curves that are
=
asymptotic to one another may sometimes have the same limit.=20
For example, The hyperbola y =3D x + 1/x has the line y=3Dx as one
of =
its two asuymptotes (the other being the line x=3D0), but x + 1/x does
=
not approach x as x grows infinite; it is said to be asymptotic to
x.
--Nelson
----- Original Message -----=20
From: Ned Granic=20
To: maths@xxxxxxxxxxxxx=20
Sent: Saturday, November 04, 2006 2:00 AM
Subject: [maths] calculus - limits
Hi all,
Apparently, I shall be the first one to ask a question on this new
=
list. What an honor!
So, here it goes:
The idea of a limit is illustrated by the secant line approaching
the =
tangent line.
Does that mean it never coincides with the tangent line at any
point =
whatsoever?
Many thanks in advance!
There is more to come.
Ned
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Ned,
No, sec A does not approach tan = A as A=20 approaches 90 degrees;
both of them grow without bound as A approaches=20 pi/2. However,
both curves are asymptotic to the vertical line at = A =3D 90=20 degrees
(as well as to each other); i.e., the distances between them =
approach=20 0.
1/x approaches 0 (in the limit) = as x grows=20 infinite; 1/x
approaches 2 in the limit as x approaches 1/2, and indeed, = it=20 equals
2 when x=3D1/2.
Don't confuse limits with = asymptotes=20 although two curves that
are asymptotic to one another may sometimes = have the=20 same limit.
For example, The hyperbola y =3D = x + 1/x has=20 the line y=3Dx
as one of its two asuymptotes (the other being the line = x=3D0), but
x=20 + 1/x does not approach x as x grows infinite; it is said to be =
asymptotic to=20 x.
--Nelson
- ----- Original Message -----
- From:=20 Ned = Granic
- To: maths@xxxxxxxxxxxxx
- Sent: Saturday, November 04, = 2006 2:00=20 AM
- Subject: [maths] calculus - = limits
- Hi all,
-
- Apparently, I shall be the first ask a=20 question on this
new list. What an honor!
- So, here it goes:
- The idea of a limit is illustrated by = the secant=20 line
approaching the tangent line.
- Does that mean it never coincides = with the=20 tangent line at any
point whatsoever?
-
- Many thanks in advance!
- There is more to come.
- Ned
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