Calculate Limit of 3/ (x (ln (x+4) - ln (x)) as x→∞

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In summary, the limit of 3/(x(ln(x+4)-ln(x))) as x approaches infinity is 0. This can be found by using L'Hospital's rule or by simplifying the expression. Finding this limit helps us understand the behavior of the function as the input values get larger, indicating that the function approaches 0 but never reaches it. This limit can also be evaluated by simplifying the expression, without using L'Hospital's rule. There are various methods for evaluating limits involving logarithmic functions, such as simplifying the expression, using L'Hospital's rule, or properties of logarithms. This limit can also be evaluated for other values of x, but it may result in a different value, as the
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Homework Statement



3/ (x (ln (x+4) - ln (x))

as x goes to infinity.

so first i look at this i i thought its going to 0 cause x is going to get larger at the bottom.

but then i see that the equation looks more like 3 / (x( ln ((x+1)/x))
ln ((x+1)/x) is going to 1. so that mean that ln 1= 0

so...
3 / x(0) = infinty?
 
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  • #2
Nope, not infinity.

Hint: ln(a) - ln(b) = ln(a/b) and (1 + a/x)x → ea as x → ∞
 
  • #3
it doesn't go to zero?
 
  • #4
No. The limit is a positive number.
 
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What is the limit of 3/(x(ln(x+4)-ln(x))) as x approaches infinity?

The limit of this function as x approaches infinity is 0. This can be found by using L'Hospital's rule or by simplifying the expression.

What is the significance of finding this limit?

Finding the limit of a function helps us understand the behavior of the function as the input values get larger. In this case, we can see that the function approaches 0 as x gets larger, indicating that the function is always getting closer to 0 but never actually reaching it.

Can this limit be evaluated without using L'Hospital's rule?

Yes, this limit can also be evaluated by simplifying the expression. By factoring out an x in the denominator and using properties of logarithms, we can see that the limit is equal to 0.

Is there a specific method for evaluating limits involving logarithmic functions?

Yes, there are several methods for evaluating limits involving logarithmic functions, such as using L'Hospital's rule, simplifying the expression, or using the properties of logarithms. The method used may depend on the complexity of the function and the desired level of precision.

Can this limit be evaluated for other values of x besides infinity?

Yes, this limit can be evaluated for any value of x, but it may result in a different value. When x is a finite number, the function will have a different behavior than when x is approaching infinity. The limit will only be equal to 0 when x approaches infinity.

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