Limits question (w/natural logs)

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In summary, as x approaches infinity, the term 2ln(x) also approaches infinity, causing the whole expression to approach 0. This is because the power of infinity in the base and the power of infinity in the exponent are not equivalent. The correct mathematical approach to finding the limit is by using the hint, which involves converting the expression to a logarithmic form.
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Melawrghk
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Homework Statement


Find limit:
limx=> inf (1+2x)11/(2ln(x))

The Attempt at a Solution


Well, I have no real mathematical attempt to this. It was just my common sense that failed. Anyways, I figured that as x goes to infinity the 2ln(x) term will also go to infinity. Since that 11 on the top doesn't change, the term will eventually go to 0. So then we'd have (1+2*infinity) to the power of zero. Which should equal one.

HOWEVER, that's wrong. I think I know why - because the infinity in the base will be different from that in the power (the power one will be increasing slower because of the ln()). But I don't know. I know there should be some mathematical way of doing this, but I don't see. Any tips?
 
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  • #2
ok i;ll give u a hint


[tex](1+2x)^\frac{11}{2ln(x)}=e^{\frac{11}{2ln(x)}ln(1+2x)}[/tex]
 

1. What is the limit of ln(x) as x approaches 0?

The limit of ln(x) as x approaches 0 is negative infinity. This is because the natural logarithm function is undefined for x=0 and approaches negative infinity as x gets closer to 0.

2. How do you find the limit of ln(x) as x approaches infinity?

The limit of ln(x) as x approaches infinity is infinity. This is because the natural logarithm function grows without bound as x increases towards infinity.

3. What is the limit of ln(x) as x approaches 1?

The limit of ln(x) as x approaches 1 is 0. This is because the natural logarithm function approaches 0 as x gets closer to 1.

4. Can you use L'Hopital's rule to find the limit of ln(x) as x approaches 0?

No, L'Hopital's rule cannot be used to find the limit of ln(x) as x approaches 0. This is because the natural logarithm function is not indeterminate at this point, as it approaches negative infinity.

5. How does the limit of ln(x) as x approaches infinity compare to the limit of ln(x) as x approaches 0?

The limit of ln(x) as x approaches infinity is infinity, while the limit of ln(x) as x approaches 0 is negative infinity. This is because the natural logarithm function behaves differently at these two points, with one approaching positive infinity and the other approaching negative infinity.

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