Department of Pre-University Education, KarnatakaPUC Karnataka Science Class 12

# Discuss the Continuity of the F(X) at the Indicated Points: F(X) = | X − 1 | + | X + 1 | at X = −1, 1. - Mathematics

Sum

Discuss the Continuity of the F(X) at the Indicated Points : F(X) = | X − 1 | + | X + 1 | at X = −1, 1.

#### Solution

Given : $f\left( x \right) = \left| x - 1 \right| + \left| x + 1 \right|$

We have
(LHL at x = −1) =  $\lim_{x \to - 1^-} f\left( x \right) = \lim_{h \to 0} f\left( - 1 - h \right)$

$= \lim_{h \to 0} \left[ \left| - 1 - h - 1 \right| + \left| - 1 - h + 1 \right| \right] = 2 + 0 = 2$
(RHL at x = −1) =
$\lim_{x \to - 1^+} f\left( x \right) = \lim_{h \to 0} f\left( - 1 + h \right)$
$= \lim_{h \to 0} \left[ \left| - 1 + h - 1 \right| + \left| - 1 + h + 1 \right| \right] = 2 + 0 = 2$
Also,
$f\left( - 1 \right) = \left| - 1 - 1 \right| + \left| - 1 + 1 \right| = \left| - 2 \right| = 2$

Now,
(LHL at x = 1) = $\lim_{x \to 1^-} f\left( x \right) = \lim_{h \to 0} f\left( 1 - h \right) = \lim_{h \to 0} \left( \left| 1 - h - 1 \right| + \left| 1 - h + 1 \right| \right) = 0 + 2 = 2$

(RHL at x =1) =  $\lim_{x \to 1^+} f\left( x \right) = \lim_{h \to 0} f\left( 1 + h \right) = \lim_{h \to 0} \left( \left| 1 + h - 1 \right| + \left| 1 + h + 1 \right| \right) = 0 + 2 = 2$

Also,

$f\left( 1 \right) = \left| 1 + 1 \right| + \left| 1 - 1 \right| = 2$
∴ ​$\lim_{x \to - 1^-} f\left( x \right) = \lim_{x \to - 1^+} f\left( x \right) = f\left( - 1 \right) and \lim_{x \to 1^-} f\left( x \right) = \lim_{x \to 1^+} f\left( x \right) = f\left( 1 \right)$
Hence,
$f\left( x \right)$   is continuous at  $x = - 1, 1$ .

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#### APPEARS IN

RD Sharma Class 12 Maths
Chapter 9 Continuity
Exercise 9.1 | Q 39.2 | Page 21