Question:
If $f(x)=|x|-3$, find $\lim _{x \rightarrow 3} f(x)$
Solution:
Left Hand Limit(L.H.L.):
$\lim _{x \rightarrow 3^{-}} f(x)$
$=\lim _{x \rightarrow 3^{-}}|x|-3$
$=\lim _{x \rightarrow 3^{-}}-(x-3)$
$=-(3-3)$
$=0$
Right Hand Limit(R.H.L.):
$\lim _{x \rightarrow 3^{+}} f(x)$
$=\lim _{x \rightarrow 3^{+}}|x|-3$
$=\lim _{x \rightarrow 3^{+}}(x-3)$
$=3-3$
$=0$
Since,
$\lim _{x \rightarrow 3^{-}} f(x)=\lim _{x \rightarrow 3^{+}} f(x)$
We can say that the limit exists and
$\lim _{x \rightarrow 3} f(x)=0$
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