Trouble Solving: The Limit of x→∞ ln(√x + 5)/ln(x)

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In summary, the conversation is about solving the limit as x approaches positive infinity of ln(square root of x + 5) divided by ln(x). The speaker is having trouble solving it and is asking for help, while another person suggests using L'hopital's rule to solve the indeterminate form.
  • #1
redsox5
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The problem is

The limit as x approaches pos infinity ln(square root of x + 5) divided by ln(x)

In the numerator only x is under the square root. I'm having trouble getting to this answer. If someone can take a look I would really appreciate it.

I have the answer because it can be done on a calculator. Just I'm not sure how to get there.
 
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  • #2
Well what is lim x-> infty ln(sqrt(x)+5) and lim x-> ln(x)?
 
  • #3
try evaluating the limit as my friend above me pointed out, you will get an indeterminate form. Fortunately L'hopitals rule can be used to deal with such limits.
 

1. What is the limit of x→∞ ln(√x + 5)/ln(x)?

The limit of x→∞ ln(√x + 5)/ln(x) is equal to 0.

2. How do you solve for the limit of x→∞ ln(√x + 5)/ln(x)?

To solve for this limit, you can use the L'Hopital's rule or the properties of logarithms to simplify the expression and then take the limit as x approaches infinity.

3. What does the limit of x→∞ ln(√x + 5)/ln(x) represent?

The limit of x→∞ ln(√x + 5)/ln(x) represents the behavior of the function as x approaches infinity. In this case, the function approaches 0 as x becomes larger and larger.

4. Can you graph the function ln(√x + 5)/ln(x)?

Yes, the function ln(√x + 5)/ln(x) can be graphed using a graphing calculator or a software such as Desmos. The graph will show that the function approaches 0 as x approaches infinity.

5. How is the limit of x→∞ ln(√x + 5)/ln(x) related to exponential growth?

The limit of x→∞ ln(√x + 5)/ln(x) is related to exponential growth because it represents the growth rate of the function as x becomes larger and larger. In this case, the function has a growth rate of 0, indicating that it approaches a horizontal asymptote and does not have exponential growth.

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