Limit of Natural Log Sequence: How to Find It Using L'Hopital's Rule?

In summary, to find the limit of the given sequence ${a}_{n} = \ln \left(\frac{12n + 2}{-9 + 4n}\right)$, divide both the numerator and denominator inside the logarithm by $n$ to get ${a}_{n} = \ln \left(\frac{12 + \frac{2}{n}}{-\frac{9}{n} + 4}\right)$. From here, you can simplify the expression and apply L'Hopital's rule to evaluate the limit.
  • #1
tmt1
234
0
I have this sequence:

$${a}_{n} = \ln \left(\frac{12n + 2}{-9 + 4n}\right)$$

I need to find the limit of this sequence. How can I go about this? Do I need to apply L'Hopitals rule? I'm unsure how to simplify this expression. If I use the rule $\ln(\frac{a}{b}) = \ln a - \ln b$ I get $\infty - \infty$, which I don't think is useful.
 
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  • #2
tmt said:
I have this sequence:

$${a}_{n} = \ln \left(\frac{12n + 2}{-9 + 4n}\right)$$

I need to find the limit of this sequence. How can I go about this? Do I need to apply L'Hopitals rule? I'm unsure how to simplify this expression. If I use the rule $\ln(\frac{a}{b}) = \ln a - \ln b$ I get $\infty - \infty$, which I don't think is useful.

Hi tmt, :)

Here's a hint. Divide both the numerator and the denominator of the fraction inside the denominator by $n$ so that you get,

$${a}_{n} = \ln \left(\frac{12 + \frac{2}{n}}{-\frac{9}{n} + 4}\right)$$

Try to continue from here. :)
 

1. What is the limit of a natural log sequence?

The limit of a natural log sequence is the value that the sequence approaches as the number of terms in the sequence increases. It is also known as the limit of ln(n) or the limit of the natural logarithm of n.

2. How is the limit of a natural log sequence calculated?

The limit of a natural log sequence can be calculated using the formula lim ln(n) = ln(L), where L is the limit of the sequence. In other words, the limit of ln(n) is equal to the natural logarithm of the limit of the sequence.

3. What is the relationship between the limit of a natural log sequence and the limit of the sequence itself?

The limit of a natural log sequence is closely related to the limit of the sequence itself. In fact, if the limit of the sequence exists, then the limit of the natural log sequence will also exist and be equal to the natural logarithm of the limit of the sequence.

4. Can the limit of a natural log sequence be infinity?

Yes, the limit of a natural log sequence can be infinity. This occurs when the limit of the sequence itself is infinity. In this case, the limit of ln(n) will also be infinity.

5. What are some real-life applications of the limit of natural log sequence?

The limit of natural log sequence has many applications in mathematics, physics, and other sciences. It can be used to calculate the growth rate of populations, the rate of change of a function, and the convergence of series. It is also important in the study of infinite sequences and series.

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