# How to make functions right-continuous

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1. Sep 9, 2016

### Sho Kano

1. The problem statement, all variables and given/known data
Given $r(t)=\left< \frac { sint }{ t } ,\frac { { e }^{ 2t }-1 }{ t } ,{ t }^{ 2 }ln(t) \right>$
Re-define $r(t)$ to make it right continuous at $t=0$

2. Relevant equations

3. The attempt at a solution
This is probably the simplest problem ever, but I don't even know what it's asking for. Right continuous as in right handed limit? How can I re-define it?

2. Sep 9, 2016

### Staff: Mentor

You need to define values for each of the three component functions so that r(0) exists, and $\lim_{t \to 0^+} r(t)$ exists and is equal to r(0).

3. Sep 9, 2016

### Sho Kano

So something like
x=1 when t=0
y=2 when t=0

4. Sep 9, 2016

### SammyS

Staff Emeritus
Correct.

How about z when t = 0 ?

5. Sep 9, 2016

### Sho Kano

z = 0 when t = 0 because 0*∞ is indeterminate if the 0 is not "constant"?

6. Sep 9, 2016

### Staff: Mentor

"Indeterminate" means you can't say what the value will be.
If you write $t^2\ln(t)$ as $\frac{\ln(t)}{t^{-2}}$, you now have the indeterminate form $[\frac{\infty}{\infty}]$, so you can use L'Hopital's Rule on it.

7. Sep 9, 2016

### Sho Kano

The limit is 0, by L'Hopital's Rule. So the way I'm re-defining it is making r(t) continuous for t is not 0, and make r(0)=<1,2,0>, like a piece-wise function

8. Sep 9, 2016

Yes