Probability Density Function

[KindofSlow]

Integral part of problem is easy.
Gets us to: Limit as u approaches infinity of -e^(-t/10) from 0 to u.
Answer is: Limit as u approaches infinity of (1-e^(-u/10)) = 1
I cannot figure out either of why either of those steps is true.
Any quidance would be greatly appreciated.
Thank you

 

[galactus]

Is this what you are getting at?:

\(\displaystyle -\lim_{u\to \infty}\int_{0}^{u}e^{\frac{-t}{10}}dt\)

First, as you done, we can integrate....getting:

\(\displaystyle \lim_{u\to \infty}\left[\frac{10}{e^{\frac{u}{10}}}-10\right]\)

\(\displaystyle 10\lim_{u\to\infty}\left[e^{\frac{-u}{10}}-1\right]\)

Now, taking the limit, we can see that \(\displaystyle e^{\frac{-u}{10}}\to 0\) as \(\displaystyle u\to \infty\)

And we are left with -10.

 

[KindofSlow]

Sorry Galactus, I left off the original integral since I don't know how to put in all the symbols.
The original integral is the integral from 0 to infinity of 0.1e^(-t/10) dt.
I'm not having any problems taking the integral.
It's the limit that I don't follow:

Limit as u approaches infinity of -e^(-t/10) from 0 to u.
= Limit as u approaches infinity of (1-e^(-u/10))
= 1

Thank you

 

[mmm4444bot]

The typing is not clear, on your first limit statement. Why did you put "from 0 to u" at the end of the limit statement?

Doing that makes it look like you're trying to take the limit of an integral, like galactus showed.

In fact, the very first line in galactus' post is a simple yes-or-no question that you have yet to answer.

So, I'm going to ignore your first limit statement.

I can explain why your second limit statement equals 1.

\(\displaystyle \lim_{u\to\infty}\left[1 - e^{-u/10}}\right]\)

Let's get rid of the negative exponent.

\(\displaystyle \lim_{u\to\infty}\left[1 - \frac{1}{e^{u/10}}}\right]\)

As u approaches infinity, it becomes a very big number.

Dividing this very big number by 10 does nothing; u/10 itself becomes infinitely larger.

This in turn means that e^(u/10) is becoming a very large power of e because the exponent keeps increasing.

In my second limit statement above, we see the ratio 1 over this growing power of e.

Think about what happens when you start dividing 1 into increasingly bazillions of pieces! The pieces become smaller and smaller (you can get them as close to nothing as you like, as shown below).

1/10 = 0.1
1/100 = 0.01
1/1000 = 0.001
1/10000 = 0.0001
1/100000 = 0.00001
1/1000000 = 0.000001
1/10000000 = 0.000001

1/100000000000000000000 = 0.00000000000000000001

1/10000000000000000000000000000000000000000 = 0.0000000000000000000000000000000000000001

Got it?

That is why the value of the ratio 1/e^(u/10) approaches zero, as u grows without bound.

In the limit, the expression 1 - 1/e^(u/10) becomes 1 - 0.