another proof in lim

[orir]

i've been told that i need to post another thread for another question, so -

f and g are function defined in a punctured neighborhood of X₀, and L is a real number.
assuming lim(x->x₀) (f times g)(x)=L

i need to prove\disprove that if lim(x->x₀) f(x)=infinity, then lim(x->x₀) g(x) exists.

 

[DrPhil]

i've been told that i need to post another thread for another question, so -

f and g are function defined in a punctured neighborhood of X₀, and L is a real number.
assuming lim(x->x₀) (f times g)(x)=L

i need to prove\disprove that if lim(x->x₀) f(x)=infinity, then lim(x->x₀) g(x) exists.

As ALWAYS, we need to see what you have tried.

Consider the properties of f(x) in the neighborhood of x_0. If the limit is +infinity, can you say in general that f(x) must behave like 1/|x - x0| ?

If f(x) has a pole but f(x)*g(x) does not, what can you say about g(x)?

 

[orir]

As ALWAYS, we need to see what you have tried.

Consider the properties of f(x) in the neighborhood of x_0. If the limit is +infinity, can you say in general that f(x) must behave like 1/|x - x0| ?

If f(x) has a pole but f(x)*g(x) does not, what can you say about g(x)?

is this enough for proving? i mean, will it be acceptable if i write only this explenation?

 

[DrPhil]

is this enough for proving? i mean, will it be acceptable if i write only this explenation?

What I was trying to do was give you some clues that you could think about. What theorems do you know about singularities and reducible singularities? How rigorous does your instructor require you to be? [Almost surely, more rigor than my general suggestions!]

 

[orir]

What I was trying to do was give you some clues that you could think about. What theorems do you know about singularities and reducible singularities? How rigorous does your instructor require you to be? [Almost surely, more rigor than my general suggestions!]

my instructor require me to give a well formal prove. but i'm kinda weak and new in writing a formal prove, so it'll be nice if you help me more (or even explain me more how to do that)

 

[orir]

anyone?

 

[HallsofIvy]

You are told that \(\displaystyle \lim_{x\to x_0} f(x)= \infty\). That means that for x close to \(\displaystyle x_0\) f(x) is very very large! What does that tell you about f(x)g(x)? What must happen to g so that f(x)g(x) is NOT very very large?