I have a question about HA, VA and minimum points

[Roni1985]

Homework Statement

For the function h(x)=\frac{x^2*e^x}{x} , which of the following are true about the graph of y=h(x)?

I. The graph has a vertical asymptote at x=0
II. The graph has a horizontal asymptote at y=0
III. The graph has a minimum point

Homework Equations

h(x)=\frac{x^2*e^x}{x}

The Attempt at a Solution

'I' is wrong because we can cancel X in the denominator and it becomes a hole (or I am wrong?), so its not an asymptote.

For 'II', I tried different ways.
My first way was putting 'ln' on each side, and trying to find the limit of that function when x goes to infinite.

For the third one, when I take the derivative, I get X=-1 when the derivative is equal to 0 and a critical point at X=0...
But I am getting a max point and not a min point.

Plus, according to my answers, 'II' and 'III' are correct.

The thing is that I haven't took cal I since the 10th grade and now I am tutoring cal I at my college, so, I need to refresh the very basic things (even though it doesn't look very basic)
Also, this question allows using a calculator, but I want to know how to solve it without.

Thanks,
Roni

 

[n!kofeyn]

I) You are correct. However, note that if you happen to define h(0)=0, which is what you get after canceling the x in the denominator, then there isn't a hole. Though either way, the function does not have an asymptote, as you said, and as stated, the function is not defined at x=0.

II) What does 'lan' mean? To find the horizontal asymptotes, if there are any, you take the limit of h(x) as x goes to infinity, and then you take the limit as x goes to -infinity. If you get a number, then y=that number is a horizontal asymptote. It is always possible to get infinity (or -infinity) for the limit, which in that case, there is no horizontal asymptote. That's why you have to check both cases.

III) There is no critical point at x=0 because the function isn't even defined there. So x=-1 is your only critical point. How did you determine the maximum? Show your methods, and check again, because you should get a maximum from that.

Also, there's a concern that you're tutoring calculus I if you couldn't figure this out on your own or with the use of a book. You should either have the book you're tutoring for, or check one out of the library. As a tutor, you should be in a position to explain and to not just being able to solve the problems by getting help of your own.

 

[Roni1985]

I) You are correct. However, note that if you happen to define h(0)=0, which is what you get after canceling the x in the denominator, then there isn't a hole. Though either way, the function does not have an asymptote, as you said, and as stated, the function is not defined at x=0.

II) What does 'lan' mean? To find the horizontal asymptotes, if there are any, you take the limit of h(x) as x goes to infinity, and then you take the limit as x goes to -infinity. If you get a number, then y=that number is a horizontal asymptote. It is always possible to get infinity (or -infinity) for the limit, which in that case, there is no horizontal asymptote. That's why you have to check both cases.

III) There is no critical point at x=0 because the function isn't even defined there. So x=-1 is your only critical point. How did you determine the maximum? Show your methods, and check again, because you should get a maximum from that.

Also, there's a concern that you're tutoring calculus I if you couldn't figure this out on your own or with the use of a book. You should either have the book you're tutoring for, or check one out of the library. As a tutor, you should be in a position to explain and to not just being able to solve the problems by getting help of your own.

A few things, for III, right, its a min point..

for II, I did to infinite and negative infinite but only with L'Hopital's rule. I don't think they are supposed to know how to use L'Hopital's rule. Anyways, this exercise allows them to use the calculator. I thought there was a simpler way to do it.

And right, as a tutor I should know how to solve all this and know how to explain it. I know that if I understand something I can explain it.
So, now I am solving final exams and that was my only problem ( though not a real problem because I could solve it with a more advanced way).
And you are right, I should open a book for the next time I face a problem :)

Thank you for your help.

 

[n!kofeyn]

A few things, for III, right, its a min point..

for II, I did to infinite and negative infinite but only with L'Hopital's rule. I don't think they are supposed to know how to use L'Hopital's rule. Anyways, this exercise allows them to use the calculator. Anyways, I thought there was a simpler way to do it.

And right, as a tutor I should know how to solve all this and know how to explain it. I know that if I understand something I can explain it.
So, now I am solving final exams and that was my only problem ( though not a real problem because I could solve it with a more advanced way).
And you are right, I should open a book for the next time I face a problem :)

Thank you for your help.

There is no reason to use l'Hopital's rule here. You can just cancel out the x in the denominator when taking the limit, so that you are taking the limit of xe^x as x goes to plus/minus infinity. This limit is infinity as x goes to positive infinity, and the limit is zero as x goes to negative infinity. You need to know why though.

There are also methods to find the minimum of such a function without looking at the graph on a calculator, so you need to review in a text the sections that cover the first and second derivative tests.

 

[Roni1985]

There is no reason to use l'Hopital's rule here. You can just cancel out the x in the denominator when taking the limit, so that you are taking the limit of xe^x as x goes to plus/minus infinity. This limit is infinity as x goes to positive infinity, and the limit is zero as x goes to negative infinity. You need to know why though.

There are also methods to find the minimum of such a function without looking at the graph on a calculator, so you need to review in a text the sections that cover the first and second derivative tests.

Yes, I found the minimum, I did something wrong and I got a max point but I found my mistake.

And when we are trying to find lim of x*e^x when it goes to negative infinite, we get negative infinite over infinite, which is undefined.

What method did you use for the limit?

 

[n!kofeyn]

Yes, I found the minimum, I did something wrong and I got a max point but I found my mistake.

And when we are trying to find lim of x*e^x when it goes to negative infinite, we get negative infinite over infinite, which is undefined.

What method did you use for the limit?

As x goes to negative infinity, the function e^x decreases (approaches zero) much faster than x goes to negative infinity. So that's why the limit is zero in that case. I guess I misspoke as you technically need to apply l'Hopital's rule here, but the reasoning above is fine.

 

[Roni1985]

As x goes to negative infinity, the function e^x decreases (approaches zero) much faster than x goes to negative infinity. So that's why the limit is zero in that case. I guess I misspoke as you technically need to apply l'Hopital's rule here, but the reasoning above is fine.

oh right, how could I forget this :\ ... lol e^x increases faster than x ...

guess I need to keep reviewing...

 

[Bohrok]

As x goes to negative infinity, the function e^x decreases (approaches zero) much faster than x goes to negative infinity. So that's why the limit is zero in that case.

How can you tell that e^x goes to 0 fast enough for that limit to be 0?

 

[n!kofeyn]

How can you tell that e^x goes to 0 fast enough for that limit to be 0?

Look at the http://www.wolframalpha.com/input/?i=graph+y=x+and+y=exp(x)" of y=e^x and y=x. You can see as x gets more and more negative, y=e^x gets closer to zero much faster than x gets more negative. The exponential function is said to dominate the limit, because it has more of an effect than the x term.

 

[Bohrok]

I meant to say without a calculator because the OP said he wanted to do it without a calculator.

 

[n!kofeyn]

I meant to say without a calculator because the OP said he wanted to do it without a calculator.

I didn't say you need a calculator. I included the graph as a reference. You should be able to graph those two functions. Also, just understand that the exponential function often dominates limits and grows much faster than x does (or decreases to zero much faster than x decreases). To prove this limit exactly, you need l'Hopital's rule as mentioned above.