Hello!

I'm having difficulty finding the limit of function x / ((x^2) - 4)) as x approaches 2 from the left (x--> 2-). My textbook tells me that the limit would be -infinite, but how does this work? With 0 in the denominator, wouldn't the limit not exist?

****Edit: I can see from graphing the function that x goes towards negative infinity as it approaches 2 from the left, but is there any way I could tell whether a limit is nonexistent/infinity without having to graph the function?

Thank you so much!

Guest Jun 6, 2022

edited by
Guest
Jun 6, 2022

#1**+1 **

Does this help?

\(\displaystyle \lim_{x\rightarrow2^-}\;\frac{x}{x^2-4}\)

If x is between 0 and 2 (not including 2) then this will be pos divided by neg which is negative

that is why this limit HAS to be negative

\(\displaystyle \lim_{x\rightarrow2^-}\;\frac{x}{x^2-4}\\~\\ =\displaystyle \lim_{x\rightarrow2^-}\;\frac{x/x^2}{(x^2-4)/x^2}\\~\\ =\displaystyle \lim_{x\rightarrow2^-}\;\frac{1/x}{1-4/x^2}\\~\\ ==> \frac{approaching \;0.5}{appraoching\; 0\; from\; negative \;side}\\~\\ =-\infty\)

Melody Jun 6, 2022