Calculus limit

Homework Statement

Hey the question is similar to this one

Evaluate the limit

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

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

rock.freak667
Homework Helper
Let $$L=\lim_{x\rightarrow \infty} (\frac{x}{x+4})^x$$

Take ln on both sides

$$ln L =ln \lim_{x\rightarrow \infty} (\frac{x}{x+4})^x \Rightarrown L =ln \lim_{x\rightarrow \infty} x ln\frac{x}{x+4}$$

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

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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.

epenguin
Homework Helper
Gold Member
Crudely
$$L=\lim_{x\rightarrow \infty} (\frac{x}{x+4})^x = \lim_{x\rightarrow \infty} (\frac{x}{x})^x = \lim_{x\rightarrow \infty}(1)^x = 1$$

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

$$\lim_{x\rightarrow \infty} (\frac{x}{x+4})^x = \lim_{x\rightarrow \infty} (\frac{x+4-4}{x+4})^x = _{x\rightarrow \infty} (\frac{x+4}{x+4} - \frac{4}{x+4})^x = etc.$$

will that do? :uhh:

tiny-tim
Homework Helper
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!

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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epenguin
Homework Helper
Gold Member
hmm … math criteria are there for a reason!

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

tiny-tim
Homework Helper
To make it obscure? :rofl:

They don't deliver in my area.

I live in a volume
… gives me room for pizza!

epenguin
Homework Helper
Gold Member
I stand corrected too.

HallsofIvy
$$L=\lim_{x\rightarrow \infty} (\frac{x}{x+4})^x = \lim_{x\rightarrow \infty} (\frac{x}{x})^x = \lim_{x\rightarrow \infty}(1)^x = 1$$
$$\lim_{x\rightarrow \infty} (\frac{x}{x+4})^x = \lim_{x\rightarrow \infty} (\frac{x+4-4}{x+4})^x = _{x\rightarrow \infty} (\frac{x+4}{x+4} - \frac{4}{x+4})^x = etc.$$