If f(x) = * 20 c |x| + 1 0 |x| - 1 . * 20 c x < 0 x = 0 x > 0 for… - Vidyadip

Question

If f(x) = $ \left\{ {\begin{array}{*{20}{c}} {|x| + 1} \\ 0 \\ {|x| - 1} \end{array}} \right.\begin{array}{*{20}{c}} {x < 0} \\ {x = 0} \\ {x > 0} \end{array}$ for what values of a does $\mathop {\lim }\limits_{x \to a}$ f(x) exists?

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Answer

Here f(x) = $ \left\{ {\begin{array}{*{20}{c}} {|x| + 1} \\ 0 \\ {|x| - 1} \end{array}} \right.\begin{array}{*{20}{c}} {x < 0} \\ {x = 0} \\ {x > 0} \end{array}$
$\Rightarrow$ f(x) = $\left\{ {\begin{array}{*{20}{c}} { - x + 1,} \\ {0,} \\ {x - 1,} \end{array}} \right.\begin{array}{*{20}{c}} {x < 0} \\ {x = 0} \\ {x > 0} \end{array}$
$\therefore \;\mathop {\lim }\limits_{x \to a}$ f(x) exists for all a $\ne$ 0
Now we see that $\mathop {\lim }\limits_{x \to 0}$ f(x) exist or not
L.H.L.= $\mathop {\lim }\limits_{x \to {0^ - }}$ f(x) = $\mathop {\lim }\limits_{x \to {0^ - }}$ (-x + 1) = 1
R.H.L =$\mathop {\lim }\limits_{x \to {0^ - }}$ f(x) = $\mathop {\lim }\limits_{x \to {0 }}$ (x - 1) = -1
$\therefore$ L.H.L at x = 0 $\ne$ R.H.L as x 0
Thus $\therefore \;\mathop {\lim }\limits_{x \to 0}$ f(x) does not exist.

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