#math/calculus The statement $ \lim_{x\to x_{0}}f(x)=L $ means that $f(x)$ can be made arbitrarily close to $L$ by taking $x$ sufficiently close—but not necessarily equal—to $x_{0}$. The value $f(x_{0})$ may differ from $L$ or may not exist at all. For example, $ f(x)=\frac{2x^{2}-7x+3}{x-3}=2x-1\quad(x\neq 3), \qquad \lim_{x\to 3}f(x)=5. $ The original formula is undefined at $x=3$, so the graph is the line $y=2x-1$ with a hole at $(3,5)$. ```tikz \begin{document} \begin{tikzpicture}[x=1cm,y=0.55cm,line cap=round] \draw[->,black!45] (-0.2,0) -- (4.8,0) node[right] {$x$}; \draw[->,black!45] (0,-0.4) -- (0,8.2) node[above] {$y$}; \draw[very thick,blue,domain=0.25:4.4] plot (\x,{2*\x-1}); \fill[white] (3,5) circle (3pt); \draw[very thick,blue] (3,5) circle (3pt); \draw[dashed,black!40] (3,0) -- (3,5) -- (0,5); \node[above left] at (3,5) {$(3,5)$}; \end{tikzpicture} \end{document} ``` The **left-hand limit** $\lim_{x\to x_{0}^{-}}f(x)$ approaches from values below $x_{0}$; the **right-hand limit** $\lim_{x\to x_{0}^{+}}f(x)$ approaches from above. A two-sided limit exists exactly when both one-sided limits exist and are equal.