Comprehension

निर्देश : निम्नलिखित प्रश्नों के लिए निम्नलिखित पर विचार करें:

माना f(x) = |x|2 - |x2| है।

निम्नलिखित कथनों पर विचार कीजिए:

I. f(x), x = 0 पर संतत है।

II. f(x), x = 1 पर संतत है।

ऊपर दिए गए कथनों में से कौन सा/से सही है/हैं?

This question was previously asked in
NDA-II 2024 (Maths) Official Paper (Held On: 01 Sept, 2024)
View all NDA Papers >
  1. केवल I
  2. केवल II
  3. I और II दोनों
  4. न तो I और न ही II

Answer (Detailed Solution Below)

Option 2 : केवल II
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Detailed Solution

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व्याख्या:

f(x) = [x]2 - [x2]

x = 0 पर

LHL = \(\rm \lim_h\rightarrow 0\) [0 - h]2 - [(0 - h)2]

= (-1)2 - 0 = 1

RHL = \(\rm \lim_h\rightarrow0 [0 + h]^2 – [(0 + h)^2] \)

= 0 - 0 = 0

LHL ≠ RHL; f(x), x = 0 पर संतत नहीं है

x = 1 पर

LHL = \(\rm \lim_h\rightarrow 0\) [[1 - h]2 - [(1 - h)2]

= 1 - 1 = 0

RHL = \(\rm \lim_h\rightarrow 0\) [1 + h]2 - [(1 + h)2]

= 1 - 1 = 0

⇒ f(x), x = 1 पर संतत है; LHL = RHL।

इसलिए, विकल्प (b) सही है।

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