Prove the following properties of limits

[A dummy progression]

Hi there are four properties to prove . Here they are:View attachment 31608

I did only the first and second properties.
Here are my attempts

View attachment 31609
View attachment 31610
Did i prove this correctly? I need your review

But the third and last properties , i didn't do them because i don't know how to use the same trick i did before. Could you help me?

Thank you for your help.

 

[Steven G]

I would have done the 1st two differently. Having said that, I like your proof.
However, your method can't be used for the last do problems.

Let y = [(ax+b)/(ax+c)]^(mx+p).
I would take ln of both side and then compute the limit.
See what you can do with this method. Even try it for the 1st two proofs.
Are you a student in Calculus 1? Are these problems in the course?

 

[Steven G]

I looked more closely at the 4th statement and don't believe it to be valid.

 

[A dummy progression]

I looked more closely at the 4th statement and don't believe it to be valid.

What is the valid one? Because this is in the book syllabi the teacher gave us

 

[A dummy progression]

I would have done the 1st two differently. Having said that, I like your proof.
However, your method can't be used for the last do problems.

Let y = [(ax+b)/(ax+c)]^(mx+p).
I would take ln of both side and then compute the limit.
See what you can do with this method. Even try it for the 1st two proofs.
Are you a student in Calculus 1? Are these problems in the course?

I'm a year 12 student in highschool