Find _ x 5 f(x), where f(x) = |x| - 5 - Vidyadip

Question

Find $\mathop {\lim }\limits_{x \to 5} f(x)$, where f(x) = |x| - 5

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Answer

Here f(x) = |x| - 5
L.H.L. $\mathop {\lim }\limits_{x \to 5} f(x) = \mathop {\lim }\limits_{x \to {5^ - }} |x| - 5$
Put x = 5 – h as x → 5, h → 0
$\therefore \;\mathop {\lim }\limits_{h \to 0} \left| {5 - h} \right| - 5 = \mathop {\lim }\limits_{h \to 0} 5 - h - 5 = \mathop {\lim }\limits_{h \to 0} ( - h) = 0$
R.H.L. $= \mathop {\lim }\limits_{x \to {5^ + }} f(x) = \mathop {\lim }\limits_{x \to {5^ + }} |x| - 5$
Put x = 5 + h as x → 5, h →0
$\therefore \;\mathop {\lim }\limits_{h \to 0} \left| {5 + h} \right| - 5 = \mathop {\lim }\limits_{h \to 0} 5 + h - 5 = \mathop {\lim }\limits_{h \to 0} h = 0$
Now L.H.L. = R.H.L
Thus limit exists at x = 5 and $\mathop {\lim }\limits_{x \to 0} f(x) = 0$

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