I'm very confused when I read some pages on the internet about limits (for functions).
Let's say I got any function f(x) given and someone tells me to find the limit (towards 3 or $\infty$ or whatever…).
I have to do that for both (+-) sides? So I show it for +3 and -3?
It's not enough if I just show it for +3?
Please tell me, this confuses me a lot and I couldn't find an answer to this.
For series, I would use one of these criterias, ratio test as an example. And there, I also don't have to show it for both sides. Or am I wrong? :S
Best Answer
The limits for $+3$ and $-3$ have nothing to do with each other. The notation you want for limits is $3^+$ and $3^-$ or $-3^+$ and $-3^-$
You are a little confused by the notation. Let's say we want to find the limit of the function for $x \to 3$. Then we need to find the limit for $x \to 3^+$ which means $x$ approaches $3$ from the right side. Then we need to find the limit for $x \to 3^-$ which means $x$ approaches $3$ from the left side. If the limits are equal then we say that $\lim_{x \to 3}$ exists.
In general, when people write $\lim_{x \to x_0} f(x)$ they mean that both
$$\lim_{x \to x_0^+} f(x)$$ - from the right side
and
$$\lim_{x \to x_0^-} f(x)$$ - from the left side
exist and are equal. Otherwise no limit $\lim_{x \to x_0} f(x)$ exists.
It doesn't matter if $x_0<0$ or $x_0>0$ at all.