One sided limit with two answers?

[Woolyabyss]

Say we have a function that is defined as y=3 except at x=2 and 5 where there are two vertical asymptotes.
would this function have a two sided limit? what if I were to take the limit when x approaches 3? would that be y=3?. what about one sided limits? If I were to take the positive limit as x approaches 2 would it just skip x = 5?

 

[DrewD]

This is not a well defined function (or at least not a clearly stated function). If $f(x)=3$ for $x\neq2$ and $x\neq5$, there cannot be vertical asymptotes at 2 and 5. But I think your question may not rely on asymptotes, but rather just values other than 3(?). How about

$f(x)=\left\{\begin{array}{ccc} 3&\mbox{for }&x\neq2,x\neq5\\ 10&\mbox{for }&x=2\\ 12&\mbox{for }&x=5\end{array}\right.$

The definition of a limit should make it clear that $\lim_{x\rightarrow c}f(x)=3$ for all $c$. In the $\epsilon$-$\delta$ definition, $0<|x-c|<\delta$ so that the value of $f(c)$ never matters (it doesn't even have to be defined). This is also true for the left and right hand limits.

Now, if you meant that the function is 3, but suddenly grows "to infinity" when near 2 or 5, then the limit wouldn't exist at 2 or 5, but neither limit depends on the other. The behavior of $f(x)$ near $x=5$ does not effect $\lim_{x\rightarrow2}f(x)$.


I think you should reread the section of your text (or rewatch a video or whatever) on limits since you seem to have a number of common misconceptions.

 

[Woolyabyss]

This is not a well defined function (or at least not a clearly stated function). If $f(x)=3$ for $x\neq2$ and $x\neq5$, there cannot be vertical asymptotes at 2 and 5. But I think your question may not rely on asymptotes, but rather just values other than 3(?). How about

$f(x)=\left\{\begin{array}{ccc} 3&\mbox{for }&x\neq2,x\neq5\\ 10&\mbox{for }&x=2\\ 12&\mbox{for }&x=5\end{array}\right.$

The definition of a limit should make it clear that $\lim_{x\rightarrow c}f(x)=3$ for all $c$. In the $\epsilon$-$\delta$ definition, $0<|x-c|<\delta$ so that the value of $f(c)$ never matters (it doesn't even have to be defined). This is also true for the left and right hand limits.

Now, if you meant that the function is 3, but suddenly grows "to infinity" when near 2 or 5, then the limit wouldn't exist at 2 or 5, but neither limit depends on the other. The behavior of $f(x)$ near $x=5$ does not effect $\lim_{x\rightarrow2}f(x)$.


I think you should reread the section of your text (or rewatch a video or whatever) on limits since you seem to have a number of common misconceptions.

I thought at x approaches 2 from the left hand side there wouldn't be a limit because it never reaches 2 also the value f(x) remains 3 as it approaches so there would be no limit?

 

[DrewD]

I thought at x approaches 2 from the left hand side there wouldn't be a limit because it never reaches 2 also the value f(x) remains 3 as it approaches so there would be no limit?

The value of $f(c)$ in the evaluation of $\lim_{x\rightarrow c}f(x)$ is never important. It doesn't even need to be defined. All that matters is the behaviour of the function around $x=c$. In this case, if $f(x)=3$ at all points around $x=2$, then the limit must be $3$ because, for all $x$ near but not equal to $x=2$ (on the left if you are interested in the lefthand limit), the function is equal to $3$.

If your text doesn't make this clear, check out Paul's Online Math Notes I think this is one of many excellent free online resources. Also, Fundamentals of Calculus is another excellent source.