Properties of limits of exponential functions

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Homework Statement
Prove the following properties (in pictures)
Relevant Equations
Using the def. Of natural logarithm
IMG_20220311_172438_515.jpg

I did only the the first three prop.
And on a means we have, on pose or posons means let there be , propriétés means properties, alors meand then.

I apologize i am a french native speaker and i am busy so i cannot rewrite this in entirely english
 

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I'm not convinced by what you have done for P1. Where did you use the properties of the natural logarithm?

In any case, you need to be clear about what you are assuming in your proof.
 
My guess is you are allowed to use
<br /> \lim _{x\to\infty} \left (1+\frac{1}{x} \right )^x = e\tag{1}.<br />
For P1, the above implies
<br /> \lim _{x\to\infty} \left (1+\frac{a}{x} \right )^x = e^{a}<br />
and therefore for any ##b\geqslant 0##
<br /> \lim _{x\to\infty} \left (1+\frac{a}{x} \right )^{bx} = e^{ba}.<br />
The others are done similarly. Your task is to manipulate the limit to look like (1).

Be careful with statements like if ##x\to 0##, then ##y:= \frac{a}{x}\to \infty\ (a>0)##. This is true assuming ##x\to 0+##. It's clear ##a,b## are some fixed constants, but specify whether they are negative/non-negative. These details are important.
 
Last edited:
PeroK said:
I'm not convinced by what you have done for P1. Where did you use the properties of the natural logarithm?

In any case, you need to be clear about what you are assuming in your proof.
I don't know how to do P4. But i proved all of the first three
 
nuuskur said:
My guess is you are allowed to use
<br /> \lim _{x\to\infty} \left (1+\frac{1}{x} \right )^x = e\tag{1}.<br />
IMG_20220311_172432_730.jpg
IMG_20220311_172438_515.jpg

For P1, the above implies
<br /> \lim _{x\to\infty} \left (1+\frac{a}{x} \right )^x = e^{a}<br />
and therefore for any ##b\geqslant 0##
<br /> \lim _{x\to\infty} \left (1+\frac{a}{x} \right )^{bx} = e^{ba}.<br />
The others are done similarly. Your task is to manipulate the limit to look like (1).

Be careful with statements like if ##x\to 0##, then ##y:= \frac{a}{x}\to \infty\ (a>0)##. This is true assuming ##x\to 0+##. It's clear ##a,b## are some fixed constants, but specify whether they are negative/non-negative. These details are important.
On pose : we set , on a : we have , alors : then.

I didn't find P4 . Could you explain it to me?
 
What is assumed of ##u## and ##v##? Details are important! Do you seek to prove
<br /> \lim _{x\to a} u(x) ^{v(x)} = \exp \left ( \lim _{x\to a} (u(x)-1)v(x) \right )?<br />
If so, then don't bother, for this is false.

It is true that
<br /> \lim _{x\to a} u(x) ^{v(x)} = u(a) ^{v(a)}<br />
assuming ##u(x)^{v(x)}## is well defined around ##a## and ##u,v## are continuous at ##a##.
 
There are two things I don't understand about this problem. First, when finding the nth root of a number, there should in theory be n solutions. However, the formula produces n+1 roots. Here is how. The first root is simply ##\left(r\right)^{\left(\frac{1}{n}\right)}##. Then you multiply this first root by n additional expressions given by the formula, as you go through k=0,1,...n-1. So you end up with n+1 roots, which cannot be correct. Let me illustrate what I mean. For this...
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