Undefined Points

**Mentallic** · Jun19-09, 09:18 AM

I am aware that for a function that is undefined at a point x=a such as

$LaTeX Code: \\underbrace{lim}_{x\\rightarrow a}f(x)=\\pm \\infty$

But it tends to infinite only because it is in the form a/0, where a $LaTeX Code: \\neq$ 0.

Undefined values in the form 0/0 can have a range of values - all reals if I'm not mistaken.

I thus set up a function f(x) multiplied by another function g(x) so that f(a)=0 and g(a) undefined. However, the functions are not in a form where they can seemingly cancel factors of the zero and undefined value.

e.g.
$LaTeX Code: h(x)=\\frac{x+1}{x^2-1}=\\frac{1}{x-1}, x\\neq \\pm 1$

So, such a function I simply came up with was

$LaTeX Code: h(x)=x*tan(x+\\frac{\\pi}{2})$

I used a graphing calculator to try understand what was happening around x=0, and it seems that

$LaTeX Code: \\underbrace{lim}_{x\\rightarrow 0}h(x)=-1$

Now I just want to understand why this limit tends to -1, not any other real values.

**arildno** · Jun19-09, 09:28 AM

Well, you might try utilizing the identity:
$LaTeX Code: tan(x+y)=\\frac{\\sin(x)\\cos(y)+\\cos(x)\\sin(y)}{\\cos (x)\\cos(y)-\\sin(x)\\sin(y)}$

**statdad** · Jun19-09, 09:40 AM

A simpler example might be

$LaTeX Code: <BR>f(x) = x \\sin\\left(\\frac 1 x \\right)<BR>$

for which

$LaTeX Code: <BR>\\lim_{x \\to 0^+} f(x) = 0<BR>$

**Mentallic** · Jun19-09, 09:43 AM

Aha

$LaTeX Code: tan(x+y)=\\frac{sin(x+y)}{cos(x+y)}$

But all I get using this result is

$LaTeX Code: tan(x+\\frac{\\pi}{2})=-cot(x)$

It isn't helping just yet.

**Mentallic** · Jun19-09, 09:51 AM

Hang on...

So the function now is $LaTeX Code: f(x)=-\\frac{x}{tan(x)}$

and since the gradients of x and tanx at x=0 are equal, this gives it the value 1?

**arildno** · Jun19-09, 10:39 AM

Indeed.

Or, as you can verify:
$LaTeX Code: x\\tan(x+\\frac{\\pi}{2})=-\\frac{x}{\\sin(x)}\\cos(x)$

**Mentallic** · Jun19-09, 11:35 PM

Well, I remember the result

$LaTeX Code: \\lim_{x \\to 0}\\frac{x}{sin(x)}=1$

and

so I guess we can deduce that:

$LaTeX Code: \\lim_{x \\to 0}-\\frac{x}{sin(x)}cos(x)=-1$

However, I'm sure that the function doesn't exist at the point x=0, so if I were to draw the function, I would leave an empty circle at the point (0,-1)?

Just like my previous mentioned function: $LaTeX Code: f(x)=\\frac{x+1}{x^2-1}$
if I were to draw this function, I would quickly notice it is the same as $LaTeX Code: f(x)=\\frac{1}{x-1}$ except $LaTeX Code: x\\neq -1$

Can I do the same for

? That is to say, can I find this equal to a simpler form (or more complicated if need be) of the same function, that instead is defined at x=0?