Proving the Limit of 1/f(x) When f(x) Goes to Infinity

In summary, Michelle showed that if the limit as x goes to a of f(x)=infinity, then lim as x goes to a of 1/(f(x) =0). This was done by proving that for any M>0 there exists an delta>0, where a<x<a+delta implies that f(x)>M. To do this, she showed that whenever 0<|x-a|<\delta, |\frac{1}{f(x)}|<\frac{1}{M}=\epsilon.
  • #1
Math_Geek
23
0
1. Homework Statement [/bProve
Prove: If the limit as x goes to a of f(x)=infinity, then lim as x goes to a of 1/(f(x) =0


Homework Equations


Need to show with a delta-epsilon proof

The Attempt at a Solution



using the definition, lim as x goes to a f(x)=infinity means that for any M>0 there exists an delta>0, where a<x<a+delta implies that f(x)>M. So using this def, I know there is M>0, there exists a delta (not sure what yet) so that a<x<a+delta and taking 1/f(x) shifts the bounds a+delta<x<a and then M would be less than or equal to 0 therefore the lim as x goes to a of 1/f(x)=0.

Am I close? Please help a girl in distress! lol
Michelle
 
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  • #2
well, what you need to show that, [tex]\lim_{x\rightarrow a}\frac{1}{f(x)}=0[/tex], in epsilon delta we have: [tex]\forall\epsilon>0,\exists\delta>0, such \ \ that \ \ |\frac{1}{f(x)}|<\epsilon [/tex].

using the fact that [tex] \lim_{x\rightarrow a}f(x)=\infty[/tex], as u stated, with a little omission, means that: For any M>0, [tex]\exists\delta>0[/tex] such that whenever

[tex]0<|x-a|<\delta[/tex] we get f(x)>M. NOw using this fact here we get that whenever

[tex]0<|x-a|<\delta[/tex] we have [tex]\frac{1}{f(x)}<\frac{1}{M}=\epsilon=>|\frac{1}{f(x)}|<\epsilon=\frac{1}{M}[/tex]

Now you only need to put everything together, because it is a little messy.

I hope this helps. But feel free to ask again, don't put too much stress on yourself!
 
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  • #3
i understand but what is [tex]? at the top, I am not sure what that is.
thanks.
 
  • #4
Math_Geek said:
i understand but what is [tex]? at the top, I am not sure what that is.
thanks.

I don't know what are u talking about? I just used latex to write those math symbols, probbably your browser, or sth, is not being able to generate those symbols, because i do not see [tex] anywhere neither at the top nor at the bottom.
 
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  • #5
that thing about shifting the bounds doesn't really make sens to me.
 
  • #6
it is something our teacher showed us how to do, but it never seems to work for me. One more question, we (meaning you) showed the limit is equal to zero becasue 1/M <epsilon? I get confused because we let m>0. so how can the limit be =0.
 
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  • #7
Math_Geek said:
it is something our teacher showed us how to do, but it never seems to work for me. One more question, we (meaning you) showed the limit is equal to zero becasue 1/M <epsilon?

We, like i wrote before on my other post: We need to show that [tex]\lim_{x\rightarrow a}\frac{1}{f(x)}=0[/tex], right?? using the fact that is in there. So using epsilon delta language, it actually means that we need to show that: [tex]\forall\epsilon>0,\exists\delta>0, such \ \ that \ \ |\frac{1}{f(x)}|<\epsilon [/tex]
Right?

Now from:
[tex] \lim_{x\rightarrow a}f(x)=\infty[/tex]
we know that for every M>0 ,[tex]\exists\delta>0[/tex] such that for every [tex]xE(a-\delta,a+\delta),[/tex] with the possible exception when x=a, we have [tex]f(x)>M[/tex], but also for that delta and for that interval we also have [tex]\frac{1}{f(x)}<\frac{1}{M}=\epsilon=>|\frac{1}{f(x )}|<\epsilon=\frac{1}{M}[/tex]

so our line at the top is fullfiled now, because we found that for every epsilon, and also for [tex]\epsilon=\frac{1}{M},\exists\delta(\epsilon)>0,[/tex] such that whenever, [tex]0<|x-a|<\delta=>|\frac{1}{f(x)}|<\frac{1}{M}=\epsilon [/tex] this is all we needed to show.
 
  • #8
got it thanks
 
  • #9
Math_Geek said:
got it
Good job!


Math_Geek said:
thanks


No, problem.
I love the idea behind this forum. It is blessed.
 
  • #10
yes it is!
 

1. What is the concept of a limit in mathematics?

A limit is a fundamental concept in calculus that describes the behavior of a function as its input approaches a certain value. It represents the value that the function approaches, or "limits" to, as the input gets closer and closer to the specified value.

2. Why is it important to prove the limit of 1/f(x) when f(x) goes to infinity?

Proving the limit of 1/f(x) when f(x) goes to infinity is important because it allows us to understand the behavior of a function as its input approaches infinity. This can help us make predictions and solve problems in various fields, such as physics, engineering, and economics.

3. How do you prove the limit of 1/f(x) when f(x) goes to infinity?

The most common method for proving the limit of 1/f(x) when f(x) goes to infinity is to use the epsilon-delta definition of a limit. This involves showing that for any positive value of epsilon, there exists a corresponding positive value of delta such that when the input to the function is within delta units of the specified value, the output is within epsilon units of the limit.

4. What are some examples of functions with a limit of 1/f(x) when f(x) goes to infinity?

Some examples of functions with a limit of 1/f(x) when f(x) goes to infinity include 1/x, 1/sin(x), and e^(-x). These functions all approach 0 as their input approaches infinity.

5. How is the concept of a limit used in real-world applications?

The concept of a limit is used in various real-world applications, such as in physics to describe the behavior of particles as they approach a certain speed or energy level, in economics to analyze the behavior of markets as prices approach a certain value, and in engineering to understand the stability and accuracy of systems as their inputs approach certain values.

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