Question

Limit as x approaches infinity of 8x[ln(x+9) - ln(x)]

Answer 1

How can I rewrite it as a fraction in order to apply L'Hopital's rule?

Answer 2

you can bring x in the denominator and write it as 1/x

Answer 3

Right but then plugging in infinity I would get infinity/0 right? And thats not defined

Answer 4

you wanted to apply LH right ?

so first differentiate numerator and denominator separately

later think about plugging it, after some simplification

Answer 5

Thats true but in order to use LH we have to first make sure that we have an indeterminate form and oo/0 isn't it

Answer 6

right,

did you try the substitution
x =1/y ?

when x-> infinity,
y->0

Answer 7

\(\large\color{slate}{\displaystyle\lim_{x \rightarrow ~\infty }\left(\ln(x+9)-\ln(x)\right)}\)

\(\large\color{slate}{\displaystyle\lim_{x \rightarrow ~\infty }\left(\ln\frac{x+9}{x}\right)}\)

\(\large\color{slate}{\displaystyle\left(\ln\left(\lim_{x \rightarrow ~\infty }\frac{x+9}{x}\right)\right)}\)

Answer 8

so you are taking the ln of that limit

Answer 9

0

Answer 10

oh 8x !!1soprry

Answer 11

Yes thats right! Oh but where's the 8x?

Answer 12

yes yes, my bad

Answer 13

And so is that also one of the limit properties? Exchanging the places of ln and limit?

Answer 14

yes limit properties allow thatr

Answer 15

Alright so using the same idea where would the 8x go?

Answer 16

its not exchanging...
Limit can be dragged inside of only those functions which are continuous

Answer 17

can you give the substitution one shot?

the substitution
x =1/y ?

when x-> infinity,
y->0