Is my homework solution correct?

[temaire]

Homework Statement

3The attempt at a solution

Is this correct?

 

[Dick]

Sure, it looks fine to me.

 

[temaire]

Do I need to show that I'm approaching 4 from the right?

 

[Dick]

Do I need to show that I'm approaching 4 from the right?

I would say you are approaching 4 from the left. t<4, correct? Why would you want to approach from the right?

 

[temaire]

I would say you are approaching 4 from the left. t<4, correct? Why would you want to approach from the right?

Because if I were approaching ln|x-4| from the right, the graph goes to -∞. How could you approach it from the left?

Or am I supposed to approach the original graph of 1/(x^2 -3x -4) from the left?

 

[Dick]

The problem with your integral is at x=4. To resolve it as an improper integral you want to integrate from x=0 to x=4-epsilon where epsilon>0. That means you are approaching the upper limit from the left. You don't care what the limit is from the right.

 

[temaire]

The problem with your integral is at x=4. To resolve it as an improper integral you want to integrate from x=0 to x=4-epsilon where epsilon>0. That means you are approaching the upper limit from the left. You don't care what the limit is from the right.

So even though the limit is approaching 4 from the left at ln|x-4|, we're infact evaluating the limit as it approaches 4 from the left of 1/(x^2-3x-4)?

 

[Dick]

So even though the limit is approaching 4 from the left at ln|x-4|, we're infact evaluating the limit as it approaches 4 from the left of 1/(x^2-3x-4)?

Now you are just confusing me. You are approaching x=4 from the left. Period. Approaching ln|x-4| from the left gives you the behavior of the integral of 1/(x^2-3x-4) on the interval [0,4].

 

[temaire]

This is what I was confused about.

In the graph above, you can approach 4 from the right but not from the left.

 

[Dick]

It's an absolute value, temaire. |x-4|. Doesn't that mean anything to you? :)

 

[temaire]

Oh, so this is the graph. (x is from 3 to 5)

I understand now.

 

[Dick]

You've got it.