• Support PF! Buy your school textbooks, materials and every day products Here!

Limit of one to the power of infinity

  • Thread starter Avichal
  • Start date
  • #1
292
0

Homework Statement


Find limit x->0 (f(x))g(x).
Here limit x->0 f(x) = 1 and limit x->0 g(x) = ∞


Homework Equations


I know that limit x->0 (1 + x)1/x = e ... (1)

The Attempt at a Solution


I have the answer which is elimit x->0 (f(x) - 1) * g(x)
But I don't know how to proceed. I have to somehow use equation (1) but how?
 

Answers and Replies

  • #2
6
0
Try writing [tex]f(x) = 1 + [f(x) - 1].[/tex] Also, note that [tex](1+u) = [(1+u)^{1/u}]^u.[/tex]
 
  • #3
292
0
Okay I got some idea. Is this right?

f(x)g(x) = ((1 + f(x) -1)1/(f(x) - 1))(f(x) - 1) * g(x)
Since (1 + f(x) -1)1/(f(x) - 1) = e ... using (1)
∴ ans = e(f(x) - 1) * g(x)
 
  • #4
arildno
Science Advisor
Homework Helper
Gold Member
Dearly Missed
9,970
132
Note that this can be written as:
[tex]e^{g\ln(f)}=e^{\frac{\ln(f)}{\frac{1}{g}}}[/tex]
The main fraction in the exponent is of L'Hopital niceness.
In theory, that is!
 
  • #5
292
0
Now that I asked this question I have another related doubt.
This isn't really related to the original question but a general limits doubt.

We know that (1 + x)1/x = e as x → 0
But why isn't (1 + x2)1/x = e as x → 0

Does the fact that x2 approaches 0 faster than x change the value of the limit? But if the power was also x2 instead of x then the limit would be e. How come?
 
  • #6
arildno
Science Advisor
Homework Helper
Gold Member
Dearly Missed
9,970
132
"But if the power was also x2 instead of x then the limit would be e. How come?"
Because it is the ratio of the RATES by which they tend to zero that is critical for what the value will be.

Look at (1+2x)^(1/x), for example. Here, x goes to zero twice as fast than 2x, meaning the limit is e^(2), rather than e
 
  • #7
292
0
"But if the power was also x2 instead of x then the limit would be e. How come?"
Because it is the ratio of the RATES by which they tend to zero that is critical for what the value will be.

Look at (1+2x)^(1/x), for example. Here, x goes to zero twice as fast than 2x, meaning the limit is e^(2), rather than e
Oh ok, yes. Thank you!

Note that this can be written as:
[tex]e^{g\ln(f)}=e^{\frac{\ln(f)}{\frac{1}{g}}}[/tex]
The main fraction in the exponent is of L'Hopital niceness.
In theory, that is!
I didn't understand the R.H.S.
Is [tex]e^{\frac{\ln(f)}{\frac{1}{g}}} = e^{f-1*g}[/tex]
 
  • #8
arildno
Science Advisor
Homework Helper
Gold Member
Dearly Missed
9,970
132
I didn't read your question properly, so unfortunately, I gave a wholly irrelevant answer. Sorry about that..
 

Related Threads on Limit of one to the power of infinity

Replies
1
Views
2K
  • Last Post
Replies
4
Views
515
  • Last Post
Replies
2
Views
929
  • Last Post
Replies
2
Views
3K
  • Last Post
Replies
3
Views
1K
  • Last Post
Replies
4
Views
1K
  • Last Post
Replies
2
Views
881
  • Last Post
Replies
3
Views
1K
  • Last Post
Replies
1
Views
3K
  • Last Post
Replies
2
Views
5K
Top