Definition
Limit
\(\lim_{x\to a^+}f(x)\): right-hand limit at \(a\)
\(\lim_{x\to a^-}f(x)\): left-hand limit at \(a\)
\(\lim_{x\to a}f(x)\): limit at \(a\)
For a function to have a limit at a point, right-hand and left-hand limits must be equal.
\[\lim_{x\to a^-}f(x)=\lim_{x\to a^+}f(x)=b\Rightarrow\lim_{x\to a}f(x)=b\]
If \(\lim_{x\to a}f(x)=f(a)\), the function is continuous at \(a\). If the two-sided limit exists but differs from \(f(a)\), the function is discontinuous at that point.
If a function is continuous, there is no need to analyze one-sided limits separately: the limit is equal to the image.
1.
\(\lim_{x\to2^+}f(x)+\lim_{x\to2^-}f(x)=?\)
2.
\(\lim_{x\to-2^+}f(x)+\lim_{x\to1^-}f(x)=?\)