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Calculus limit

  • Thread starter oven_man
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  • #1
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Homework Statement



Hey the question is similar to this one

Evaluate the limit

lim (x/x+4)^x
x->infinity


Homework Equations





The Attempt at a Solution



my attempt was

to change it to

lim x->infinity e^(x.(ln(x/x+4))
then i dont know where to go from there
 

Answers and Replies

  • #2
rock.freak667
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Let [tex]L=\lim_{x\rightarrow \infty} (\frac{x}{x+4})^x[/tex]

Take ln on both sides

[tex]ln L =ln \lim_{x\rightarrow \infty} (\frac{x}{x+4})^x \Rightarrown L =ln \lim_{x\rightarrow \infty} x ln\frac{x}{x+4}[/tex]
 
  • #3
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Here is a hint:
[tex]\lim_{x\rightarrow \infty} (1 + \frac{c}{x})^x = e^c[/tex], where c is a constant.
 
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  • #4
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Well, Latex is not co-operating, but rock.freak667's comment just needs a little tweaking. There's an extra L in the second part of the equation, and an extra 'ln' in the 3rd part, and the 'x' should be moved to the front. Since I tried altering what he's posted by quoting, and consequently fell flat on my face with Latex difficulties, it's almost certain that rock.freak suffered from the troubles.
 
  • #5
epenguin
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Crudely
[tex]L=\lim_{x\rightarrow \infty} (\frac{x}{x+4})^x = \lim_{x\rightarrow \infty} (\frac{x}{x})^x = \lim_{x\rightarrow \infty}(1)^x = 1[/tex]

Personally I do these things in intuitive way; it may not satisfy math criteria you are supposed to. :wink: For me it has the advantage that the answer is fairly obvious, I suppose I could be misled in some way in some strange cases. Although that was sufficient for me it might be better to insert a step

[tex]\lim_{x\rightarrow \infty} (\frac{x}{x+4})^x = \lim_{x\rightarrow \infty} (\frac{x+4-4}{x+4})^x = _{x\rightarrow \infty} (\frac{x+4}{x+4} - \frac{4}{x+4})^x = etc.[/tex]

will that do? :uhh:
 
  • #6
tiny-tim
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Personally I do these things in intuitive way; it may not satisfy math criteria you are supposed to.
hmm … math criteria are there for a reason!

Stick with pizzasky's method! :smile:
 
  • #7
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I'm not sure... Since (x/ (x+4)) is a touch under 1, putting it to the power of infinity will reduce it towards zero (but not necessarily zero). Just try this with x=100, x=1000, x=10^6. The drop off is apparent.
 
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  • #8
epenguin
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hmm … math criteria are there for a reason!
To make it obscure? :rofl:

Stick with pizzasky's method! :smile:
They don't deliver in my area. :smile:
 
  • #9
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I stand corrected. Pizzasky's method (which is correct) shows e^-4 is the limit, and this agrees with the number I arrived at using the computer's calculator, which found e^-3.999999992 as the number when x = 10^9. well done
 
  • #10
tiny-tim
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To make it obscure? :rofl:
Nooo … to help you pass the exams! :wink:
They don't deliver in my area. :smile:
I live in a volume
:biggrin: … gives me room for pizza! :biggrin:
 
  • #11
epenguin
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I stand corrected too.:redface:
 
  • #12
HallsofIvy
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Crudely
[tex]L=\lim_{x\rightarrow \infty} (\frac{x}{x+4})^x = \lim_{x\rightarrow \infty} (\frac{x}{x})^x = \lim_{x\rightarrow \infty}(1)^x = 1[/tex]

Personally I do these things in intuitive way; it may not satisfy math criteria you are supposed to. :wink: For me it has the advantage that the answer is fairly obvious, I suppose I could be misled in some way in some strange cases. Although that was sufficient for me it might be better to insert a step

[tex]\lim_{x\rightarrow \infty} (\frac{x}{x+4})^x = \lim_{x\rightarrow \infty} (\frac{x+4-4}{x+4})^x = _{x\rightarrow \infty} (\frac{x+4}{x+4} - \frac{4}{x+4})^x = etc.[/tex]

will that do? :uhh:
To make it obscure? :rofl:
How about: To get the right answer. The limit here is NOT 1!
 

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