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]

\(\displaystyle \lim_{x \to c^-}\) approaches c from the negative side of the x-axis and \(\displaystyle \lim_{x \to c^+}\) approaches c from the positive side. For example \(\displaystyle \lim_{x \to 0^+} \dfrac{1}{x} \to + \infty\) whereas \(\displaystyle \lim_{x \to 0^-} \dfrac{1}{x} \to - \infty\).

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

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