Limit as x approaches Infinity

In summary: And that is just wrong. Try reading the other posts, they did mention "e" several times for a good reason.
  • #1
caesius
24
0

Homework Statement


Evaluate [tex]lim (1+a/x)^x[/tex]
(that's limit as x tends to +infinity, sorry don't know how to latex that)

Homework Equations


The Attempt at a Solution


Stumped. Having that [tex]x[/tex] exponent has my confused. As x tends to infinity I know what's inside the brackets will tend to one, but the exponent will make it tend to infinity. But the question has three marks to it, it can't be that simple.
 
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  • #2
so
call your limit
f(a)
-show the the limit exists (the a=1 case should be familar)
-show f(x)f(y)=f(x+y)
-show f is continuous and differentiable at some point
now you should be able to identify f
 
  • #3
Caesius, have you seen all definitions of e?

Oh and your latex that you wanted to do is \lim_{x \rightarrow \infty} which becomes [tex]\lim_{x \rightarrow \infty}[/tex]
 
  • #4
[tex]\lim_{x\rightarrow\infty}\left(1+\frac a x\right)^x[/tex]

Set it equal to y, take the natural log of both sides.

[tex]y=\left(1+\frac a x\right)^x[/tex]

[tex]\lim_{x\rightarrow\infty}\ln y=\lim_{x\rightarrow\infty}x\ln\left(1+\frac a x\right)[/tex]

[tex]\lim_{x\rightarrow\infty}\ln y=\lim_{x\rightarrow\infty}x\ln\left(1+\frac a x\right)[/tex]

Now you have an Indeterminate form of [tex]\infty\cdot0[/tex]

Keep solving till you can apply L'Hopital's Rule on the right side, then you will have "solved for [tex]\ln y[/tex], so use algebra to solve for "y" and you're pretty much done.
 
  • #5
DavidWhitbeck said:
Caesius, have you seen all definitions of e?
QUOTE]

No one has seen all definitions of e as there are an infinite number of them.
 
  • #6
rocomath said:
[tex]\lim_{x\rightarrow\infty}\left(1+\frac a x\right)^x[/tex]

Set it equal to y, take the natural log of both sides.

[tex]y=\left(1+\frac a x\right)^x[/tex]

[tex]\lim_{x\rightarrow\infty}\ln y=\lim_{x\rightarrow\infty}x\ln\left(1+\frac a x\right)[/tex]

[tex]\lim_{x\rightarrow\infty}\ln y=\lim_{x\rightarrow\infty}x\ln\left(1+\frac a x\right)[/tex]

Now you have an Indeterminate form of [tex]\infty\cdot0[/tex]

Keep solving till you can apply L'Hopital's Rule on the right side, then you will have "solved for [tex]\ln y[/tex], so use algebra to solve for "y" and you're pretty much done.


Thats a great solution but I think no need to go that long;

as x approaches infinity "(a/x)" equates 0 .

Hence : 1+0 = 1
 
  • #7
mubashirmansoor said:
Thats a great solution but I think no need to go that long;

as x approaches infinity "(a/x)" equates 0 .

Hence : 1+0 = 1

And that is just wrong. Try reading the other posts, they did mention "e" several times for a good reason.
 
  • #8
mubashirmansoor said:
Thats a great solution but I think no need to go that long;

as x approaches infinity "(a/x)" equates 0 .

Hence : 1+0 = 1
Indeterminate Power of form: [tex]1^{\infty}[/tex] so yes it is necessary.
 

1. What does "limit as x approaches Infinity" mean?

The phrase "limit as x approaches Infinity" refers to the behavior of a function as the input value, x, gets larger and larger without bound. In other words, it describes what happens to the output of a function as the input approaches infinity.

2. How do you calculate the limit as x approaches Infinity?

To calculate the limit as x approaches Infinity, you can use the properties of limits and evaluate the function at higher and higher values of x. If the function approaches a specific value or approaches infinity, then that value is the limit as x approaches Infinity.

3. What does it mean if the limit as x approaches Infinity equals zero?

If the limit as x approaches Infinity equals zero, it means that the function approaches zero as the input value gets larger and larger without bound. This suggests that the function is approaching the x-axis and getting closer to zero, but it never actually reaches zero.

4. What is the difference between a horizontal asymptote and the limit as x approaches Infinity?

A horizontal asymptote is a line that a function approaches as the input value gets larger without bound. The limit as x approaches Infinity is the value that the function is approaching as the input value gets larger without bound. In some cases, the horizontal asymptote and the limit as x approaches Infinity may be the same value, but this is not always the case.

5. Can the limit as x approaches Infinity be a negative value?

Yes, the limit as x approaches Infinity can be a negative value. This means that the function is approaching a negative value as the input value gets larger without bound. It is important to note that the limit as x approaches Infinity can be a positive, negative, or undefined value, depending on the behavior of the function.

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