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 Pre-Calculus and Calculus
 Limit to infinity
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alexdc
Junior Member

USA
5 Posts

Posted - 07/10/2011 :  19:45:36  Show Profile  Reply with Quote
Hi,

I have a question involving a limit to infinity that I am unsure of how to solve. I have the answer, and will post my method for solving the problem. Though I am not sure if I have gone about solving this problem in the correct manner. I would value some feedback on this. Thank you.

Problem:

limit of

t * ( (t + 1) - t)

as

t --> infinity

Answer:

infinity

My work:

Begin by multiplying by the conjugate:

((t + 1) + t) / ( (t + 1) + t)

to get:


t * ( (t + 1 - t) / ((t + 1 + t) )


Simplified:

t * ( 1 / ((t + 1) + t) )

Taken to infinity:

t --> infinity

1 --> 1

(t + 1) + t --> infinity

1 / ((t + 1) + t) --> 0

Now here is where I'm confused. I understand that the answer to this question is infinity. I understand intuitively why this works. MATHEMATICALLY, however, I am confused. How is it that we determine that when one portion of this equation approaches 0, and the other approaches infinity, that, when multiplied together, these two portions become inifinity? I know that the equation:

1 / ((t + 1) + t)

Only approaches 0, without ever really becoming zero. But how am I supposed to prove mathematically that:

t * ( 1 / ((t + 1) + t) )

approaches infinity, rather than zero? Have I solved this problem wrong, maybe?

Edited by - alexdc on 07/10/2011 19:48:21
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royhaas
Moderator

USA
3059 Posts

Posted - 07/11/2011 :  09:20:50  Show Profile  Reply with Quote
Go back to the statement after "Simplified:". You can't take limits like that' in other words, you can't take the limit of something going to infinity and multiply it by something that is not going to infinity.But you could rewrite the expression as

2 t/( (t+1)+t ) t /2.

Can you take it from there?
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alexdc
Junior Member

USA
5 Posts

Posted - 07/11/2011 :  21:28:10  Show Profile  Reply with Quote
Never mind, I understand what I did wrong. Starting with the expression:

t * ( 1 / ((t + 1) + t) )

I write this as:

t / ((t + 1) + t)

and mulitply both this equation by:

( 1 / t ) / ( 1 / t )

to get:

( t / t ) / ( ((t + 1) + t) / t )

Simplified, step one:

t / ( ((t + 1) / t ) + (t / t) )

Simplified, step two:

t / ( ( (t / t) + (1 / t) ) + (t / t) )

Simplified, step three:

t / ( ( 1 + (1 / t) ) + 1 )

which yields:

1/t --> 0

1 --> 1

( 1 + (1 / t) ) --> 1

1 --> 1

( ( 1 + (1 / t) ) + 1 ) --> 2

t --> infinity

The limit of the equation is then:

( ( ( 1 + (1 / t) ) + 1 ) ) --> inifinity / 2

and:

infinity / 2 --> inifinity
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