# $1^\infty$-Limit Cases

This is to prove that if $\displaystyle \lim_{x \to \infty} f(x) = 1$ and $\displaystyle \lim_{x \to \infty} g(x) = \infty$, then $\displaystyle \lim_{x \to \infty} f(x)^{g(x)} = e^{\lim_{x \to \infty} g(x)(f(x) - 1)}$.

\begin{aligned} \lim_{x \to \infty} f(x)^{g(x)} & = \lim_{x \to \infty} \left(1+f(x)-1\right)^{g(x)} \\ & = \lim_{x \to \infty} \left(1+\frac 1{\frac 1{f(x)-1}} \right)^{g(x)\left(\frac {f(x)-1}{f(x)-1}\right)} \\ & = \lim_{x \to \infty} \left[\left(1+\frac 1{\frac 1{f(x)-1}} \right)^\frac 1{f(x)-1}\right]^{g(x)(f(x)-1)} \\ & = \lim_{x \to \infty} e^{g(x)(f(x)-1)} \\ & = e^{\lim_{x \to \infty} g(x)(f(x)-1)} \end{aligned}

Note by Chew-Seong Cheong
1 year, 3 months ago

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Hello Sir! How are you? Remember me? I got admitted to an IIT this year!

- 12 months ago

Yes, how are you?

- 12 months ago

I am fine sir, How are you?

- 11 months ago

- 11 months ago

IIT bhu right

- 7 months, 1 week ago

Yeah! Do i know you?

- 6 months, 3 weeks ago

No,bhaiya i was on your group on quora

- 6 months, 3 weeks ago

Okay bro nice! Which class you are in?

- 6 months, 3 weeks ago

Bhaiya,I am in 10 th class.Can you give me some tips regarding boards and prmo.....and should i learn computer languages or just focus on boards???Please reply.

- 6 months, 3 weeks ago

Boards mein kuch ni rkha h bro. Thoda padhlio obviously, kyuki aas pados wale puchte hai. Baaki computer language seekh le,agar accha lgta hai, continue with that. It will help a lot in future.

- 6 months, 3 weeks ago

Aur bhaiya prmo ke liye........

- 6 months, 3 weeks ago

you contact me through quora, im inactive here kriti.

- 4 months, 4 weeks ago

Ok bhaiya!

- 4 months, 4 weeks ago

Zuhair is cool :)

- 2 months, 3 weeks ago

Bravo! You are awesome, Chew-Seong Cheong. I have seen your solutions all over Brilliant! I could use your help with a couple of mathematical mysteries if you are willing. I will start with one concerning (x'(t))^2+(y'(t))^2=1. (I did the work for this by myself, so there could be a mistake.) Please respond when you have time, and I will convey the trouble I am having with this topic. Again, if you are willing...

- 2 days, 12 hours ago