Stuck on limit question (limit of e): [e^{a/x} + 1] / [e^{b/x} + 1] = a + b

[SirJobz]

Hello, so I have a problem in this limit question

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\(\displaystyle \large{\displaystyle \mbox{Giving }\, a,\, b\, >\, 0\, \mbox{ and }\, \lim_{x \rightarrow 0^+}\, \left( \dfrac{e^{\frac{a}{x}}\, +\, 1}{e^{\frac{b}{x}}\, +\, 1} \right) \, =\, a\, +\, b,\, \mbox{ find }\, a\, +\, b.}\)

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so I'd tried to solve this problem by using L'Hospital,
but it's still not getting the answer.

So what should I do and what is the real answer to this and how?
Thank you.

 

[Dr.Peterson]

Hello, so I have a problem in this limit question

---

\(\displaystyle \large{\displaystyle \mbox{Giving }\, a,\, b\, >\, 0\, \mbox{ and }\, \lim_{x \rightarrow 0^+}\, \left( \dfrac{e^{\frac{a}{x}}\, +\, 1}{e^{\frac{b}{x}}\, +\, 1} \right) \, =\, a\, +\, b,\, \mbox{ find }\, a\, +\, b.}\)

---

so I'd tried to solve this problem by using L'Hospital,
but it's still not getting the answer.

So what should I do and what is the real answer to this and how?
Thank you.

Please show your work, so we can see whether you made a mistake, or just didn't go far enough. How do you know you are not "getting the answer"? (Do you get an answer that someone else says is wrong, or are you unable to get an answer at all, or what?)

 

[SirJobz]

So I think I’d made the mistake because in question they said a,b > 0 and you can find a and b value.
that’s how I stuck in this question.

 

[Dr.Peterson]

View attachment 10187
So I think I’d made the mistake because in question they said a,b > 0 and you can find a and b value.
that’s how I stuck in this question.

What is the derivative of a/x? (Not of ax!)