Asymptotes In Pre-Calculus

I am currently learning about the asymptotes of rational functions in precalculus. However, 4 things about it confuse me, specifically about the asymptotes of it, and how to calculate them.

Why is it when given a rational function with 2 polynomials of the same degree on their numerators and denominators, the asymptote can be found by dividing the coefficients of the highest degree terms?

Why is it if the polynomial in the numerator is a lower degree than the denominator, the x-axis (y = 0) is the horizontal asymptote?

Why is it if the polynomial in the numerator is a higher degree than the denominator, there is no horizontal asymptote?

Why is it when the polynomial in the numerator is a higher degree than the polynomial in the denominator, there will be a slant asymptote which you can find through polynomial long division?

Can you explain all of this using simple algebra, without complicated techniques such as Euler's division of polynomials? I don't understand any complicated techniques and theorems beyond the quadratic formula. Can you also show and explain your working, so it is easier for me to follow through? I am still a beginner, so that would help very much

I am currently learning about the asymptotes of rational functions in precalculus. However, 4 things about it confuse me, specifically about the asymptotes of it, and how to calculate them.

Why is it when given a rational function with 2 polynomials of the same degree on their numerators and denominators, the asymptote can be found by dividing the coefficients of the highest degree terms?

Why is it if the polynomial in the numerator is a lower degree than the denominator, the x-axis (y = 0) is the horizontal asymptote?

Why is it if the polynomial in the numerator is a higher degree than the denominator, there is no horizontal asymptote?

Why is it when the polynomial in the numerator is a higher degree than the polynomial in the denominator, there will be a slant asymptote which you can find through polynomial long division?

Can you explain all of this using simple algebra, without complicated techniques such as Euler's division of polynomials? I don't understand any complicated techniques and theorems beyond the quadratic formula. Can you also show and explain your working, so it is easier for me to follow through? I am still a beginner, so that would help very much