limits

mathshelpplease

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Hello, I have been struggling with this problem and got this far, but I am not sure how to proceed in actually solving to show that f(x) = 0
Any help would be much appreciated.

Consider the function:
[math]f(x)=(1-1/4x)exp(-x^2+5x-6)[/math]Show the following holds:
Screenshot 2021-10-27 at 22.25.17.png

I have calculated that
[math]f'(x)=((2x^2-13x+19)exp(-x^2+5x-6))/4[/math]and from that, that if f(x) = 0, [math]x=(13+sqrt17)/4[/math]or [math]x=(13-sqrt17)/4[/math]
but am unsure how to show that this means that x tends to infinity
 

Jomo

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If you want to find when f(x) = 0, why are you computing the derivative?
If you meant you want to solve for f'(x)=0, then why are you trying to do this?
Have you learned L'Hopital's rule? Try it.
 

mathshelpplease

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that would make a lot more sense thank you!

for L'Hopital's rule would

Screenshot 2021-10-27 at 23.23.39.png[math](1-1/4x)exp(-x^2+5x-6)[/math] would then equalScreenshot 2021-10-27 at 23.23.39.png[math]((2x^2-13x+19)exp(-x^2+5x-6))/4[/math]
or have I made an error since its not f(x)/g(x) and f'(x)/g'(x)
 

mathshelpplease

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Scratching that, I think I have it worked out.

Instead, I changed [math]exp(-x^2+5x-6)[/math] to [math]exp(x^2+5x+6)[/math], putting it as the denominator of the fraction with [math](1-x/4)[/math] on top.

I then differentiated them separately once, and then again to get
[math]0/(4x^2-20x+27)exp(x^2-5x+6)[/math] which is equal to 0.

Does this answer the question? I'm sorry I have never used the L'Hopital rule so just want to make sure I haven't missed anything obvious!
 

Jomo

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What is the derivative of 1-x/4?
What is the derivative of e^(x^2-5x+6)?
What do you get when you divide them?
 

Jomo

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I then differentiated them separately once, and then again to get
[math]0/(4x^2-20x+27)exp(x^2-5x+6)[/math] which is equal to 0.
How did you get 0 in the numerator? Why did you differentiate twice? Answer my questions above so we can find your error.
 

lookagain

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The limit in the post is [imath]\displaystyle \mathop {\lim }\limits_{x \to \infty } \left( {1 - \frac{1}{{4x}}} \right)\exp ( - {x^2} + 5x - 6)[/imath]

\(\displaystyle \ (1 - 1/4x) \ \) is the same as \(\displaystyle \ (1 - \tfrac{1}{4}x) \ \) because of the Order of Operations. I would
not write it that way as the former, because I want to emphasize the fraction. In
horizontal style I would write it as \(\displaystyle \ \) (1 - (1/4)x) \(\displaystyle \ \) or \(\displaystyle \ \) [1 - (1/4)x].

For a corresponding example, if you enter "1 - 1/4x = 0" into WolframAlpha or other
linear equation solving sites where the display shows it as a horizontal style, you'll
see the solution is x = 4, but not x = 1/4.

To get \(\displaystyle \ \bigg(1 - \frac{1}{4x} \bigg), \ \) I would expect that to be indicated in horizontal style using most likely parentheses such as this: \(\displaystyle \ \ \) (1 - 1/(4x))\(\displaystyle \ \) or \(\displaystyle \ \) [1 - 1/(4x)], because the denominator
of 4x needs the grouping symbols in that case.
 
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mathshelpplease

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How did you get 0 in the numerator? Why did you differentiate twice? Answer my questions above so we can find your error.
I changed [math]exp(5x-x^2-6)[/math] to [math]exp(-(5x-x^2-6))[/math] to place it as the denominator following this example

Screenshot 2021-10-28 at 08.35.42.png
from https://calcworkshop.com/derivatives/lhopitals-rule/

I then differentiated both to get

[math](-1/4)/(2x-5)exp(x^2-5x+6)[/math]
I thought I would have to differentiate again? but is the limit now determinate as [math](-1/4)/((2x-5)exp(x^2-5x+6) = 0/infinity[/math]
differentiating again I was following
Screenshot 2021-10-28 at 08.39.29.png
In an attempt to prove directly that f(x) = 0 when x tends to infinity
 

Jomo

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Since you changed the problem you need to see if you need to even use L'Hopital and whether or not you are allowed to even use it.
As x->oo, what is the numerator approaching?
As x->oo, what is the denominator approaching?
 

mathshelpplease

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Since you changed the problem you need to see if you need to even use L'Hopital and whether or not you are allowed to even use it.
As x->oo, what is the numerator approaching?
As x->oo, what is the denominator approaching?
I am still quite lost with this problem, I have never worked with limits before.

These are my current workings out, which I think support the use of L'Hopital since g'(x) does not = 0, both are differentiable, and the limits are indetermined?
Screenshot 2021-10-28 at 17.00.05.png
 

Jomo

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The very end say infinity/infinity = 0. Why do you think that infinity/infinity = 0? Also, why is -1/9 approaching infinity as x goes anywhere?
 

mathshelpplease

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Thank you for your reply, would it then be that as x goes to infinity -1/9 would go to zero, meaning it would be zero/infinity, and therefore zero?
 

Dr.Peterson

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Thank you for your reply, would it then be that as x goes to infinity -1/9 would go to zero, meaning it would be zero/infinity, and therefore zero?
Not quite. -1/9 is a constant, and doesn't go anywhere! Its limit is -1/9.

So if you divide a negative constant by a function that goes to infinity, what is the limit?
 

mathshelpplease

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Not quite. -1/9 is a constant, and doesn't go anywhere! Its limit is -1/9.

So if you divide a negative constant by a function that goes to infinity, what is the limit?
Thank you, would it then also be 0?, I know from the question that f(x)=0 as lim x-> infinity
 
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