$1^\infty$-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 $\displaystyle \lim_{x \to \infty} f(x)^{g(x)} = e^{\lim_{x \to \infty} g(x)(f(x) - 1)}$ .

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

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Comments

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

Md Zuhair - 1 year, 4 months ago

Yes, how are you?

Chew-Seong Cheong - 1 year, 4 months ago

I am fine sir, How are you?

Md Zuhair - 1 year, 3 months ago

@Md Zuhair – Fine as well. If you want we can add me on Facebook https://www.facebook.com/chewseong.cheong

Chew-Seong Cheong - 1 year, 3 months ago

IIT bhu right

A Former Brilliant Member - 1 year ago

Yeah! Do i know you?

Md Zuhair - 11 months, 2 weeks ago

No,bhaiya i was on your group on quora

A Former Brilliant Member - 11 months, 2 weeks ago

@A Former Brilliant Member – Okay bro nice! Which class you are in?

Md Zuhair - 11 months, 2 weeks ago

@Md Zuhair – 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.

A Former Brilliant Member - 11 months, 2 weeks ago

@A Former Brilliant Member – 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.

Md Zuhair - 11 months, 2 weeks ago

@Md Zuhair – Aur bhaiya prmo ke liye........

A Former Brilliant Member - 11 months, 2 weeks ago

@A Former Brilliant Member – you contact me through quora, im inactive here kriti.

Md Zuhair - 9 months, 4 weeks ago

@Md Zuhair – Ok bhaiya!

A Former Brilliant Member - 9 months, 3 weeks ago

@A Former Brilliant Member – Zuhair is cool :)

Md Zuhair - 7 months, 2 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...

James Wilson - 5 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

\(1^\infty\)-Limit Cases

Comments