Properties of limits of exponential functions

[A dummy progression]

Homework Statement

Prove the following properties (in pictures)

Relevant Equations

Using the def. Of natural logarithm

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

 

[PeroK]

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.

 

[nuuskur]

My guess is you are allowed to use
$$\lim _{x\to\infty} \left (1+\frac{1}{x} \right )^x = e\tag{1}.$$
For P1, the above implies
$$\lim _{x\to\infty} \left (1+\frac{a}{x} \right )^x = e^{a}$$
and therefore for any $b\geqslant 0$
$$\lim _{x\to\infty} \left (1+\frac{a}{x} \right )^{bx} = e^{ba}.$$
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.

 

[A dummy progression]

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

 

[A dummy progression]

My guess is you are allowed to use
$$\lim _{x\to\infty} \left (1+\frac{1}{x} \right )^x = e\tag{1}.$$

For P1, the above implies
$$\lim _{x\to\infty} \left (1+\frac{a}{x} \right )^x = e^{a}$$
and therefore for any $b\geqslant 0$
$$\lim _{x\to\infty} \left (1+\frac{a}{x} \right )^{bx} = e^{ba}.$$
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?

 

[nuuskur]

What is assumed of $u$ and $v$? Details are important! Do you seek to prove
$$\lim _{x\to a} u(x) ^{v(x)} = \exp \left ( \lim _{x\to a} (u(x)-1)v(x) \right )?$$
If so, then don't bother, for this is false.

It is true that
$$\lim _{x\to a} u(x) ^{v(x)} = u(a) ^{v(a)}$$
assuming $u(x)^{v(x)}$ is well defined around $a$ and $u,v$ are continuous at $a$.