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How do you explain left and right hand limits in calculus?
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Our definition of the limit requires that the function get arbitrarily close to the limit value when approaching from both the left and right hand sides. However, we can evaluate limits by specifying that we only approach from one side. We usually notate this with a superscript $+$ or $-$ next to the $x$ limit value. So,
$$\lim _{x \rightarrow 0^{+}} f(x)$$
would mean "the limit of $f(x)$ as $x$ approaches 0 from the right", while
$$\lim _{x \rightarrow 0^{-}} f(x)$$
would mean "the limit of $f(x)$ as $x$ approaches 0 from the left."

Our definition of the limit from both sides requires the left and right sides to be the same. If they are different, the the limit does not exist.
$$\lim _{x \rightarrow c^{+}} f(x)=\lim _{x \rightarrow c^{-}} f(x) \Longrightarrow \lim _{x \rightarrow c^{+}} f(x)=\lim _{x \rightarrow c^{-}} f(x)=\lim _{x \rightarrow c} f(x)$$
$\lim _{x \rightarrow c^{+}} f(x) \neq \lim _{x \rightarrow c^{-}} f(x) \Longrightarrow \lim _{x \rightarrow c} f(x)$ does not exist (DNE).
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