Homework Help: Calculus limit

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]

 

Here is a hint:
[tex]\lim_{x\rightarrow \infty} (1 + \frac{c}{x})^x = e^c[/tex], where c is a constant.

 

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.

 

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. 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:

 

hmm … math criteria are there for a reason!

Stick with pizzasky's method!

 

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.

 

To make it obscure? :rofl:

They don't deliver in my area.

 

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

 

Nooo … to help you pass the exams!

I live in a volume …

… gives me room for pizza! ​

 

I stand corrected too.

 

How about: To get the right answer. The limit here is NOT 1!