Question

Calculus: Comparison Test & Limit Comparison Test

https://tinyurl.com/cntwxsb

Right after it says, "Thus the limit in question
is equal to the limit of the function" why is x → 0 instead of x → ∞

Answer 1

becasue as n to inf, an to 0

let an = x, and do x to 0

Answer 2

put x=1/n, you usually get that. what is your original question?

Answer 3

I don't understand why x → 0 but I know it has something to do with lim an as n → ∞

Answer 4

\[\lim_{n\to inf}\frac{ln(1+a_n)}{a_n}\]given that a_n to 0 as n to inf; let a_n = x
\[\lim_{x\to 0}\frac{ln(1+x)}{x}\]

Answer 5

so, as n to inf, an to 0. but since x= an, as n to inf, x to 0

Answer 6

or rather :)

\[\lim_{n\to inf}\frac{ln(1+a_n)}{a_n}~:~a_n\to0\]
\[\lim_{a_n\to 0}\frac{ln(1+a_n)}{a_n}~:~x=a_n\]
\[\lim_{x\to 0}\frac{ln(1+x)}{x}\]
its just a matter of substitutions along the way to clean tings up

Answer 7

Aha! Now I understand it :)

Answer 8

at higher levels, they assume youve been taught well enough to see what theyve done. as opposed to the lower levels where every little adding and subtrcting is spelled out in detail