Solving a Limit: x→∞ (x/x+4)^x

[oven_man]

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 don't know where to go from there

 

[rock.freak667]

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

Take ln on both sides

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

 

[pizzasky]

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

 

[BrendanH]

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]

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?

 

[tiny-tim]

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!

 

[BrendanH]

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.

 

[epenguin]

hmm … math criteria are there for a reason!

To make it obscure?

Stick with pizzasky's method!

They don't deliver in my area.

 

[BrendanH]

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]

To make it obscure?

Nooo … to help you pass the exams!

They don't deliver in my area.

I live in a volume …

… gives me room for pizza! ​

 

[epenguin]

I stand corrected too.

 

[HallsofIvy]

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?

To make it obscure?

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