Need help Proving infinite limit properties.

In summary: They say that since limx->c f(x)= infinity, it means that for every N>0 you can find a \delta_1>0 so that for every x that satisfied |x-c|<\delta_1, we get that |f(x)|>N. Similarly, since limx->c g(x)=L, for every \epsilon>0, we can find a \delta_2>0 so that for every x that satisfied |x-c|<\delta_2, we get that |g(x)-L|<\epsilon.Now, using these two inequalities, they can manipulate things so that they get a lower bound on |f(x)g(x)|. For example, they show that
  • #1
leoxy520
2
0
Suppose that limx->c f(x) = infinity and limx->c g(x)=l where l is a real number. Prove the following.
limx->c[f(x)+g(x)]= infinity
limx->c[f(x)g(x)]= infinity if l > 0
limx->c[f(x)g(x)]= -infinity if l < 0

I have the proof for these already, but I couldn't understand them, would someone please explain them.
The thing I don't understand is where does the L come from, but explanation in general would be greatly appreciated

Here is the proof for the second one:

[URL=http://imageshack.us/photo/my-images/190/unledmkd.png/][PLAIN]http://img190.imageshack.us/img190/2665/unledmkd.png[/URL]

Uploaded with ImageShack.us[/PLAIN]
 
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  • #2
There seems to be a change of notation. In the original, g -> c when x -> a. In the equations in question, g -> L when x -> c.
 
  • #3
mathman said:
There seems to be a change of notation. In the original, g -> c when x -> a. In the equations in question, g -> L when x -> c.

sorry, i corrected it, does anyone have the explanation for this proof ?
 
  • #4
What do you mean "where does the L come from"? It is the limit of the function g(x).

I remind you, what you need to prove, is that given an M>0, you can find a [itex]\delta>0[/itex], so that for every x that satisfied |x-c|<[itex]\delta[/itex],
|f(x)g(x)| > M hold.

The proof shows the existence of such a [itex]\delta>0[/itex].

In order to do this alone, however, you should start from the end. You should ask yourself, how can I make |f(x)g(x)| > M happen?
You need to remember that f(x) can be as big as you want when you're close enough to c, and that g(x) can be as close to L as you'd like when you're close enough to c.

That's exactly what they used in the proof.
 

Related to Need help Proving infinite limit properties.

1. What are the basic properties of infinite limits?

Some basic properties of infinite limits include:
- If the limit of f(x) as x approaches a is infinity, then the limit of 1/f(x) as x approaches a is 0.
- If the limit of f(x) as x approaches a is infinity and the limit of g(x) as x approaches a is infinity, then the limit of f(x) + g(x) as x approaches a is infinity.
- If the limit of f(x) as x approaches a is infinity and the limit of g(x) as x approaches a is c (a finite number), then the limit of f(x) + g(x) as x approaches a is also infinity.
- If the limit of f(x) as x approaches a is infinity and the limit of g(x) as x approaches a is infinity, then the limit of f(x) * g(x) as x approaches a is infinity.
- If the limit of f(x) as x approaches a is infinity and the limit of g(x) as x approaches a is c (a finite number), then the limit of f(x) * g(x) as x approaches a is also infinity.

2. How do I prove infinite limit properties?

To prove infinite limit properties, you can use the definition of a limit and apply it to the specific property you are trying to prove. This involves showing that for any given value of epsilon, you can find a corresponding value of delta such that the limit expression holds true. You can also use algebraic manipulations and known limit properties to simplify the expression and arrive at the desired result.

3. Can infinite limit properties be used to evaluate limits of rational functions?

Yes, infinite limit properties can be used to evaluate limits of rational functions. This is because rational functions are composed of polynomial functions, which have well-defined limits as x approaches infinity. By applying the properties of infinite limits, you can simplify the rational function and evaluate the limit by plugging in the appropriate values.

4. Are there any special cases where infinite limit properties do not apply?

Yes, there are some special cases where infinite limit properties do not apply. One example is when dealing with oscillating or undefined functions such as sin(x)/x or 1/x. In these cases, you would need to use other techniques such as L'Hopital's rule or the Squeeze theorem to evaluate the limit.

5. How do I know which infinite limit property to use?

To determine which infinite limit property to use, you need to carefully examine the given limit expression and see which property can be applied. This involves identifying any common factors, simplifying the expression, and determining what happens to the terms as x approaches infinity. It may also be helpful to graph the function or use a table of values to gain a better understanding of the behavior of the function as x approaches infinity.

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