Finding the Ordered Pairs for an Exponential Limit

In summary, the conversation is about finding the number of ordered pairs of (a,b) that satisfy a given exponential limit equation. The solution involves using L'Hopital's rule, but a mistake was made in the differentiation process. The correct solution simplifies to ab = 6.
  • #1
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Homework Statement


If ##a, b \in \{1,2,3,4,5,6\}##, then number of ordered pairs of ##(a,b)## such that ##\lim_{x\to0}{\left(\dfrac{a^x + b^x}{2}\right)}^{\frac{2}{x}} = 6## is

Homework Equations

The Attempt at a Solution


So, this is a typical exponential limit.

##\lim_{x\to0}e^{\frac{2}{x}.\ln\left(\frac{a^x + b^x}{2}\right)} = 6##

Using L'Hospital

##\lim_{x\to0}e^{2.\frac{2}{a^x + b^x}.(a^x\ln a + b^x\ln b)} = 6##

This on substituting the limit simplifies to

##e^{2.(\ln a + \ln b)} = 6##

##e^{\ln {(ab)}^2} = 6 \Rightarrow {(ab)}^2 = 6 ## However, the answer only has ##ab = 6##. What's wrong?
 
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  • #2
You can't use L'Hopital's rule there!

That said, I see you're using it inside a continuous function, which I guess is a generalisation of L'Hopital. You just made a simple error with your differentiation.
 
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  • #3
Oh, yesss, I see it now. That was rather dumb. Thank you!
 

1. What is an exponential limit?

An exponential limit is a value that a function approaches as the input (x-value) approaches infinity. It is often represented as a horizontal asymptote on a graph.

2. How do I find the ordered pairs for an exponential limit?

To find the ordered pairs for an exponential limit, you can use the function's rule to plug in values for x and solve for y. As x approaches infinity, the values for y will approach the exponential limit.

3. Can I use a calculator to find the ordered pairs for an exponential limit?

Yes, you can use a calculator to find the ordered pairs for an exponential limit. Make sure to use a graphing calculator and input the function's rule to see the graph and determine the ordered pairs.

4. What is the significance of finding the ordered pairs for an exponential limit?

Finding the ordered pairs for an exponential limit can help us understand the behavior of a function as x approaches infinity. It can also help us identify any horizontal asymptotes on the graph.

5. Are there any tricks or shortcuts to finding the ordered pairs for an exponential limit?

There are no specific tricks or shortcuts for finding the ordered pairs for an exponential limit. It is important to understand the concept and use the function's rule to plug in values for x and solve for y. Practice and familiarity with exponential functions can also make the process easier.

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