A community for students.
Here's the question you clicked on:
 0 viewing
cody_123
 11 months ago
\[\lim_{x \rightarrow \infty} \frac{ x^4*\sin \frac{ 1 }{ x }+x^2 }{ 1+\left x \right^3}\]
cody_123
 11 months ago
\[\lim_{x \rightarrow \infty} \frac{ x^4*\sin \frac{ 1 }{ x }+x^2 }{ 1+\left x \right^3}\]

This Question is Closed

mathslover
 11 months ago
Best ResponseYou've already chosen the best response.0Yep.. working on it. Did you try using (a^3 + b^3) identity? not sure whether that will work or not.

myininaya
 11 months ago
Best ResponseYou've already chosen the best response.1on that x is that a cube? I can't read for some reason. I can click on the code.

mathslover
 11 months ago
Best ResponseYou've already chosen the best response.0I suggest you simplify the denominator... use the identity.

cody_123
 11 months ago
Best ResponseYou've already chosen the best response.0that mod x will open with minus?

ganeshie8
 11 months ago
Best ResponseYou've already chosen the best response.1may be divide numerator and denominator by x^3

myininaya
 11 months ago
Best ResponseYou've already chosen the best response.1so since x<0 then x=x and since we have x^3 then we could replace x^3 with (x)^3=x^3 now recall that if u>0 then sin(u)/u>1 see if you can use that here divide both top and bottom by 1/x

myininaya
 11 months ago
Best ResponseYou've already chosen the best response.1the some l'hopital could be used :)

Zarkon
 11 months ago
Best ResponseYou've already chosen the best response.1for large x \[\Large \frac{ x^4\sin \frac{ 1 }{ x }+x^2 }{ 1+\left x \right^3}\approx \frac{ x^4\sin \frac{ 1 }{ x }+x^2 }{ \left x \right^3}\] \[\Large=\frac{ x^4\sin \frac{ 1 }{ x }+x^2 }{ x^2\left x \right}\] \[\Large=\frac{ x^2\sin \frac{ 1 }{ x }+1 }{ \left x \right}\] \[\Large\approx\frac{ x^2\sin \frac{ 1 }{ x } }{ \left x \right}=\frac{x}{x}x\sin\frac{1}{x}\]

cody_123
 11 months ago
Best ResponseYou've already chosen the best response.0we have to write x=1/h and then h>0

Zarkon
 11 months ago
Best ResponseYou've already chosen the best response.1as \(x\to \infty\) you then get \((1)\cdot 1=1\)

Zarkon
 11 months ago
Best ResponseYou've already chosen the best response.1x to infinity you get what i have above

cody_123
 11 months ago
Best ResponseYou've already chosen the best response.0@Zarkon how did u use the approximation in the first step?

Zarkon
 11 months ago
Best ResponseYou've already chosen the best response.1in my first step I got rid of the 1 since for large x the 1 really contributes almost nothing

Zarkon
 11 months ago
Best ResponseYou've already chosen the best response.1and by large x I mean large negative (obviously)

myininaya
 11 months ago
Best ResponseYou've already chosen the best response.1I would have done it like this: (but this is because i'm not commander data like zarkon (who knows all because he is a superior being to a human) \[\lim_{x \rightarrow  \infty}\frac{\frac{x^4 \sin(\frac{1}{x})}{\frac{1}{x}}+\frac{x^2}{\frac{1}{x}}}{\frac{1}{\frac{1}{x}}\frac{x^3}{\frac{1}{x}}}\] then use the fact that if u>0 then sin(u)/u>1 it should be pretty easy after this point though.

sidsiddhartha
 11 months ago
Best ResponseYou've already chosen the best response.3as xtends to  infinity mod(x)=x now try to put \[x=\frac{ 1 }{ t}\]

cody_123
 11 months ago
Best ResponseYou've already chosen the best response.0i was thinking that only

myininaya
 11 months ago
Best ResponseYou've already chosen the best response.1or i guess you could have just said \[\lim_{x \rightarrow \infty}\frac{x^4 \sin(\frac{1}{x})+x^2}{1x^3}=\lim_{x \rightarrow \infty}\frac{x^3 \frac{\sin(\frac{1}{x})}{\frac{1}{x}}+x^2}{1x^3}\]

sidsiddhartha
 11 months ago
Best ResponseYou've already chosen the best response.3dw:1402332193100:dw

sidsiddhartha
 11 months ago
Best ResponseYou've already chosen the best response.3do u get it ? @cody_123

cody_123
 11 months ago
Best ResponseYou've already chosen the best response.0can u write the intermediate steps too? thanks

sidsiddhartha
 11 months ago
Best ResponseYou've already chosen the best response.3dw:1402332508279:dw now just substitute x=(1/t)

cody_123
 11 months ago
Best ResponseYou've already chosen the best response.0\[\lim_{x \rightarrow \infty} \frac{ (\frac{ 1 }{ t })^4*\sin(t)+(\frac{ 1 }{ t^2 }) }{1+ \frac{1 }{ t^3 } }\]

cody_123
 11 months ago
Best ResponseYou've already chosen the best response.0is it correct @sidsiddhartha ?

sidsiddhartha
 11 months ago
Best ResponseYou've already chosen the best response.3no x=1/t so dw:1402332969397:dw it will be like this

cody_123
 11 months ago
Best ResponseYou've already chosen the best response.0ok @sidsiddhartha then ?

sidsiddhartha
 11 months ago
Best ResponseYou've already chosen the best response.3now just try to simplify it little more

cody_123
 11 months ago
Best ResponseYou've already chosen the best response.0\[\frac{ \frac{ 1 }{ t }*\sin t+t}{ 1+t^3 }\]

sidsiddhartha
 11 months ago
Best ResponseYou've already chosen the best response.3ya good now can do it :)

cody_123
 11 months ago
Best ResponseYou've already chosen the best response.0w8 a second in denominator i should be 1t^3

sidsiddhartha
 11 months ago
Best ResponseYou've already chosen the best response.3nope its all right now just use limit t tends to 0 (sint/t)=1

cody_123
 11 months ago
Best ResponseYou've already chosen the best response.0\[1+\frac{ 1 }{ t^3 }=\frac{ t^31 }{ t^3 }\]

cody_123
 11 months ago
Best ResponseYou've already chosen the best response.0so t^31 will come in denominator?

sidsiddhartha
 11 months ago
Best ResponseYou've already chosen the best response.3dw:1402334075049:dw its not ""

cody_123
 11 months ago
Best ResponseYou've already chosen the best response.0\[\frac{ \frac{ 1 }{ t }*\sin t+t }{ 1+t^2 }\]

sidsiddhartha
 11 months ago
Best ResponseYou've already chosen the best response.3it should be (t^3+1)

cody_123
 11 months ago
Best ResponseYou've already chosen the best response.0ya in the denominator

cody_123
 11 months ago
Best ResponseYou've already chosen the best response.0\[\frac{ t1 }{ t^3+1 }\]
Ask your own question
Sign UpFind more explanations on OpenStudy
Your question is ready. Sign up for free to start getting answers.
spraguer
(Moderator)
5
→ View Detailed Profile
is replying to Can someone tell me what button the professor is hitting...
23
 Teamwork 19 Teammate
 Problem Solving 19 Hero
 Engagement 19 Mad Hatter
 You have blocked this person.
 ✔ You're a fan Checking fan status...
Thanks for being so helpful in mathematics. If you are getting quality help, make sure you spread the word about OpenStudy.