Find the value

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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