# Limit problem

## Homework Statement

If Lim f(x) and Lim g(x) both exist and are equal
x→a x→a then Lim[f(x)g(x)]=1
x→a

## Homework Equations

No relevant equations are required in this problem. To determine whether the statement is true or false [/B]

## The Attempt at a Solution

The statement is false but the reason behind it is quite unclear to me. Is it because when the limit approaches a from left and right the limits are -∞ and +∞?? Would be very helpful if anyone of you could explain it.

## Answers and Replies

BvU
Science Advisor
Homework Helper
Hard to read your post. Something like

If ##\displaystyle \lim_{x\rightarrow a} f(x)## and ##\displaystyle \lim_{x\rightarrow a} g(x)## both exist and are equal ##\Rightarrow \displaystyle \lim_{x\rightarrow a} f(x)g(x) = 1 ## ?

Could that be ##\displaystyle \lim_{x\rightarrow a} f(x)/g(x) = 1 ## ?

• To me ##\displaystyle \lim_{x\rightarrow a}## means "x approaching from the left to a" so I wouldn't worry about "coming from the right"
• To me ##\displaystyle \lim_{x\rightarrow a} f(x) = \pm \infty## means the limit does not exist, so I wouldn't worry about those either
• But the division cause a problem in a particular case, that can serve as a counter-example that makes the general statement not true,

Last edited:
Ray Vickson
Mark44
Mentor
Hard to read your post. Something like

If ##\displaystyle \lim_{x\rightarrow a} f(x)## and ##\displaystyle \lim_{x\rightarrow a} g(x)## both exist and are equal ##\Rightarrow \displaystyle \lim_{x\rightarrow a} f(x)g(x) = 1 ## ?

Could that be ##\displaystyle \lim_{x\rightarrow a} f(x)/g(x) = 1 ## ?

• To me ##\displaystyle \lim_{x\rightarrow a}## means "x approaching from the left to a" so I wouldn't worry about "coming from the right"
The notation ##\lim_{x \to a} f(x)## indicates a two-sided limit in which x can approach a from either side. If the two-sided limit exists, then both one-sided limits also exist. Maybe that's what you were trying to say, but what you actually said wasn't clear.
BvU said:
• To me ##\displaystyle \lim_{x\rightarrow a} fx) = \pm \infty## means the limit does not exist, so I wouldn't worry about those either
• But the division cause a problem in a particular case, that can serve as a counter-example that makes the general statement not true,
I agree that the first post was unclear, and that the OP meant division instead of multiplication.

BvU
Ray Vickson
Science Advisor
Homework Helper
Dearly Missed
Hard to read your post. Something like

If ##\displaystyle \lim_{x\rightarrow a} f(x)## and ##\displaystyle \lim_{x\rightarrow a} g(x)## both exist and are equal ##\Rightarrow \displaystyle \lim_{x\rightarrow a} f(x)g(x) = 1 ## ?

Could that be ##\displaystyle \lim_{x\rightarrow a} f(x)/g(x) = 1 ## ?

• To me ##\displaystyle \lim_{x\rightarrow a}## means "x approaching from the left to a" so I wouldn't worry about "coming from the right"
• To me ##\displaystyle \lim_{x\rightarrow a} fx) = \pm \infty## means the limit does not exist, so I wouldn't worry about those either
• But the division cause a problem in a particular case, that can serve as a counter-example that makes the general statement not true,

Standard usage is that ##x \to a## means a two-sided limit (##|x-a| \to 0##). Left-hand limits are typically denoted as ##x \to a\!-## or ##x \to a\!-\!0## or ##x \uparrow a##. Right-hand limits are typically denoted as ##x \to a\!+## or ##x \to a\!+\!0## or ##x \downarrow a##.

BvU
BvU
Science Advisor
Homework Helper
Good thing you corrected me, gentlemen. Been too long ago, but I do recognize the ##x\downarrow a## and ##x\uparrow a\;.\ ## Never had much opportunity to make good use of the distinction. Apologies to professor D.