MHB Limit Laws and Techniques: What is the difference between left and right limits?

[Tracy18]

What is the difference of x->c^- and x->c^+?

 

[topsquark]

[math]\lim_{x \to c^-}[/math] approaches c from the negative side of the x-axis and [math]\lim_{x \to c^+}[/math] approaches c from the positive side. For example [math]\lim_{x \to 0^+} \dfrac{1}{x} \to + \infty[/math] whereas [math]\lim_{x \to 0^-} \dfrac{1}{x} \to - \infty[/math].

-Dan

 

[HOI]

Or you can have a "piecewise" function like
f(x)= 3x+ 1 if x< 2 and
f(x)= 2x- 2 if x> 2

Then [math]\lim_{x\to 2^-} f(x)= \lim_{x\to 2} 3x+ 1= 3(2)+ 1= 7[/math] since we only look at x less than 2 and
[math]\lim_{x\to 2^+} f(x)= \lim_{x\to 2} 2x- 2= 2(2)- 2= 2[/math] since we only look at x greater than 2.

For a general function, g, the limit, [math]\lim_{x\to a} g(x)[/math] exists if and only if both [math]\lim_{x\to 2^-} g(x)[/math] and [math]\lim_{x\to a+} g(x)[/math] exist and are equal (and, of course, [math]\lim_{x\to a} g(x)[/math] is their common value).