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

# Discuss the Continuity of the F(X) at the Indicated Points: (I) F(X) = | X | + | X − 1 | at X = 0, 1. - Mathematics

Discuss the continuity of the f(x) at the indicated points:

(i) f(x) = | x | + | x − 1 | at x = 0, 1.

#### Solution

Given:

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

We have
(LHL at x = 0) =

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

(RHL at x = 0) =

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

Also

$f\left( 0 \right) = \left| 0 \right| + \left| 0 - 1 \right| = 0 + 1 = 1$

Now,

(LHL at x = 1) =

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

(RHL at x =1) =

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

Also,

$f\left( 1 \right) = \left| 1 \right| + \left| 1 - 1 \right| = 1 + 0 = 1$
$f\left( 1 \right) = \left| 1 \right| + \left| 1 - 1 \right| = 1 + 0 = 1$
$\lim_{x \to 0^-} f\left( x \right) = \lim_{x \to 0^+} f\left( x \right) = f\left( 0 \right) and \lim_{x \to 1^-} f\left( x \right) = \lim_{x \to 1^+} f\left( x \right) = f\left( 1 \right)$

Hence,

$f\left( x \right)$  is continuous at
$x = 0, 1$
Is there an error in this question or solution?

#### APPEARS IN

RD Sharma Class 12 Maths
Chapter 9 Continuity
Exercise 9.1 | Q 39.1 | Page 21
The text passage contains a mathematical question about the continuity of a function at specific points, and it provides a detailed solution to this question.
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    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

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TYPE: <one of the following values>
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- MULTIPLE_QUESTIONS
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Department of Pre-University Education, KarnatakaPUC Karnataka Science Class 12

# Discuss the Continuity of the F(X) at the Indicated Points: (I) F(X) = | X | + | X − 1 | at X = 0, 1. - Mathematics

Discuss the continuity of the f(x) at the indicated points:

(i) f(x) = | x | + | x − 1 | at x = 0, 1.

#### Solution

Given:

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

We have
(LHL at x = 0) =

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

(RHL at x = 0) =

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

Also

$f\left( 0 \right) = \left| 0 \right| + \left| 0 - 1 \right| = 0 + 1 = 1$

Now,

(LHL at x = 1) =

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

(RHL at x =1) =

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

Also,

$f\left( 1 \right) = \left| 1 \right| + \left| 1 - 1 \right| = 1 + 0 = 1$
$f\left( 1 \right) = \left| 1 \right| + \left| 1 - 1 \right| = 1 + 0 = 1$
$\lim_{x \to 0^-} f\left( x \right) = \lim_{x \to 0^+} f\left( x \right) = f\left( 0 \right) and \lim_{x \to 1^-} f\left( x \right) = \lim_{x \to 1^+} f\left( x \right) = f\left( 1 \right)$

Hence,

$f\left( x \right)$  is continuous at
$x = 0, 1$
Is there an error in this question or solution?

#### APPEARS IN

RD Sharma Class 12 Maths
Chapter 9 Continuity
Exercise 9.1 | Q 39.1 | Page 21
user
The text passage contains a mathematical question about the continuity of a function at specific points, and it provides a detailed solution to this question.

TYPE: SINGLE_QUESTION_WITH_ANSWER