Limit of logarithmic functions

In summary, the limit of ln(4^i-1) over ln(2^i) goes to 2, which can be proven using L'Hopital's rule or by reasoning that ln(4^i-1) behaves similarly to ln(4^i) for large values of i.
  • #1
rostbrot
12
0
[tex]lim_{i\rightharpoonup\infty}[/tex] [tex]\frac{ln(4^{i}-1)}{ln(2^{i})}[/tex]

If I set this up right it should go to 2, but I'm pretty rusty and every time I try to work this out I end up getting garbage or repeating behaviors that I can't do anything with... Anyone know what exactly to do with it?

edit:
Ack, since this isn't homework I posted it here, but since it's such a basic level could someone move it to the calculus homework forum? Sorry guys :/.
 
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  • #2
Have u tried L'Hopitals Rule?
 
  • #3
and it goes to 2. another way to reason about it is, that ln(4^i-1) for i large behaves similarly with ln(4^i), so it means you can replace one with the other. THen using logarithmic rules you get as a rezult 2.
 
  • #4
...I must be missing something...

This is what I've been doing:
http://img684.imageshack.us/img684/3658/loglimj.jpg [Broken]

After taking the derivative and simplifying it down I end up with a similar case to what I had before (lines 3 and 9).
 
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  • #5
You can't rearrange a limit problem into a product and differentiate using the product rule; you need to differentiate the numerator and denominator separately
[tex]\lim_{x\rightarrow c}\frac{f(x)}{g(x)} = \lim_{x\rightarrow c}\frac{f'(x)}{g'(x)}[/tex] (when f(c) and g(c) make the form 0/0 or ∞/∞)
 
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  • #6
Wow, I can't believe I completely forgot how to use L'Hopital's rule properly...
Thanks guys!
 

1. What is the limit of a logarithmic function as x approaches infinity?

The limit of a logarithmic function as x approaches infinity is equal to infinity. This means that the logarithmic function increases without bound as x gets larger and larger.

2. How do you find the limit of a logarithmic function algebraically?

To find the limit of a logarithmic function algebraically, you can use the properties of logarithms to simplify the function and then evaluate the limit. Alternatively, you can use L'Hopital's rule if the function is in an indeterminate form.

3. Can the limit of a logarithmic function be negative?

Yes, the limit of a logarithmic function can be negative. This occurs when the function approaches negative infinity as x approaches a certain value.

4. What is the difference between the limit of a logarithmic function and the value of the function at a specific point?

The limit of a logarithmic function represents the behavior of the function as x approaches a certain value, while the value of the function at a specific point represents the output of the function at that particular input.

5. Are there any restrictions on the domain of a logarithmic function when finding its limit?

Yes, there are restrictions on the domain of a logarithmic function when finding its limit. The input of the logarithmic function must be strictly greater than 0 in order for the limit to exist.

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