Evaluating Limit: x to Infinity

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In summary, the conversation discusses evaluating the limit as x tends to positive infinity of the natural logarithm of the expression (1 + 1/sqrt(2))^x + (1 - 1/sqrt(2))^x divided by x. One approach is to use the binomial series expansion, but it becomes complicated. Another approach is to isolate the important term and use properties of logarithms to simplify the expression. Ultimately, the limit evaluates to ln(1 + 1/sqrt(2)).
  • #1
haoku
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I am having difficulty in evaluating the limit
x tends to positive infinity (log( (1+[sqrt(2)/2])^x + (1-[sqrt(2)/2])^x ))/x

I have tried using binomial series expansion but turn out to be something messy.

Any ideas on it?
:)
 
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  • #2
Well.

You have
[tex]\lim_{x\to \infty} \frac{ \log_e \left( \left( 1 + \frac{1}{\sqrt{2}} \right)^x + \left( 1 - \frac{1}{\sqrt{2}} \right)^x \right)}{x}[/tex]

As x tends to infinity, the second term inside the natural log goes to zero, as so we can exclude that from our limit so that it easily evaluates to [tex]\log_e \left( 1 + \frac{1}{\sqrt{2}} \right)[/tex].

This gives us the value of the limit, but really is just a heuristic argument. For some rigor, further investigation is required. Guided by our previous evaluation of the limit, we can see that it would be helpful to isolate the important term as such:

[tex]\log_e \left( \left( 1 + \frac{1}{\sqrt{2}} \right)^x + \left( 1 - \frac{1}{\sqrt{2}} \right)^x \right) = \log_e \left( 1 + \frac{1}{\sqrt{2}} \right)^x + \log_e \left( 1 + (3-2\sqrt{2})^x \right)[/tex] using log ab = log a + log b.

One can easily show the second term goes to zero, using a Taylor expansion if required.
 
  • #3
Greetings:

The previous poster was indeed correct in that the numerator's second term goes to zero. That said, the limit becomes,

limit(x-->inf) [(ln(1 + 1/sqrt(2))^x) / x]. From the property log(u^n) = n*log(u), we have,

limit [x*ln(1 + 1/sqrt(2)) / x] = limit [ln(1 + 1/sqrt(2))] = ln(1 + 1/sqrt(2)) [limit of a constant is the constant].

Regards,

Rich B.
rmath4u2@aol.com
 

What is a limit?

A limit is the value that a function or sequence approaches as its input or index increases or decreases without bound. It is also known as the "end behavior" of a function or sequence.

Why do we need to evaluate limits?

Evaluating limits helps us understand the behavior of a function or sequence as its input or index approaches a specific value, such as infinity. It also helps us determine the continuity and differentiability of a function.

What is the process for evaluating a limit as x approaches infinity?

The process for evaluating a limit as x approaches infinity involves simplifying the function and then plugging in increasingly large values for x to see if there is a limit or if the function approaches infinity.

Can a limit as x approaches infinity have a finite value?

Yes, a limit as x approaches infinity can have a finite value if the function approaches a specific value or approaches a specific value from both positive and negative directions. This is known as a one-sided or two-sided limit, respectively.

How can evaluating limits as x approaches infinity help in real-world applications?

Evaluating limits as x approaches infinity can help in real-world applications by predicting the long-term behavior of a system or process. It can also be used to model and analyze exponential growth or decay, such as in population growth or radioactive decay.

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