Question

Limit,

When it says to show for any integer n what does it mean ? http://screencast.com/t/VQyTJhdwg

Answer 1

like how can I "show " ?

Answer 2

usually by induction... have you learned L'Hopital's rule? Have you learned about series?

Answer 3

l hopital yes , series no , I am basically learning alone

Answer 4

use can use L'Hopital's...
the first:
\[\lim_{x \rightarrow +\infty}\frac{ \ln x }{ x^n }\overset{H}{=}\lim_{x \rightarrow +\infty}\frac{ \frac{ 1 }{ x } }{ n\cdot x^{n-1} }=\lim_{x \rightarrow +\infty}\frac{ 1 }{ n\cdot x^{n} } = 0\text{, }\forall n \in \mathbb{Z}^+\]

Answer 5

do the same for the other... no induction needed.

Answer 6

ps. it's for any positive integer n, not any integer.

Answer 7

do you know what this says?

Answer 8

i mean, what the limit says?

Answer 9

b) results in inf/0 , is that allowed or am I doing something wrong ? Wolfram says that inf/0 is "complex infinity" ... isn't that undefined ?

Answer 10

no, you should get infinity/1

Answer 11

:o

Answer 12

how did you do it ? can you show me please ?

Answer 13

look at what i did previously and just flip it. remember, when you get 1/x to move the x to the appropriate place (numerator or denominator) and don't leave it as 1/x.

Answer 14

you should get \[\lim_{x \rightarrow +\infty}\frac{ n\cdot x^{n} }{ 1 }\]

Answer 15

look. it gives me inf * inf

Answer 16

x n x^(n-1) = n x^n
when put x = infinity
nx^n = infinity

Answer 17

really? you're not so comfortable moving things around and you should be at this level!\[\lim_{x \rightarrow +\infty}\frac{ x^n }{ \ln x }\overset{H}{=}\lim_{x \rightarrow +\infty}\frac{ n\cdot x^{n-1} }{ \frac{ 1 }{ x } }=\lim_{x \rightarrow +\infty}\frac {n\cdot x^{n-1} }{ 1} \cdot \frac{ x }{ 1 } =\lim_{x \rightarrow +\infty}\frac {n\cdot x^{n} }{ 1} = +\infty \text{, }\forall n \in \mathbb{Z}^+\]

Answer 18

Oh I see , thanks

Answer 19

You are right bro, I should see this coming , proof that I need some practice !

Answer 20

thanks btw

Answer 21

it's okay... I've taught calc before and most students struggle with the algebra and trig. experience is a great teacher, too. so do your best to remember how to make something look different even though it's really the same. kind of like getting dressed up for halloween