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নিম্নলিখিত অপেক্ষকগুলির মধ্যে কোনটি (0, 1) ব্যবধানে অভিন্নভাবে অবিচ্ছিন্ন?

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CSIR UGC (NET) Mathematical Science: Held On (7 June 2023)
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  1. f(x) = sin\(\rm\frac{1}{x}\)
  2. f(x) = e−1/x2
  3. f(x) = excos\(\rm\frac{1}{x}\)
  4. f(x) = cos x cos\(\rm\frac{\pi}{x}\)

Answer (Detailed Solution Below)

Option 2 : f(x) = e−1/x2
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ধারণা:

একটি অপেক্ষক y = f(x) একটি মুক্ত ব্যবধান (a, b) এ অভিন্নভাবে অবিচ্ছিন্ন হয় যদি f(x) (a, b) এ অবিচ্ছিন্ন হয় এবং শেষ বিন্দু a, b তে সীমা বিদ্যমান থাকে।

ব্যাখ্যা:

(1): f(x) = sin\(\rm\frac{1}{x}\)

\(\lim_{x\to0}\sin\frac1x\) বিদ্যমান নেই তাই f(x) = sin\(\rm\frac{1}{x}\) (0, 1) এ অভিন্নভাবে অবিচ্ছিন্ন নয়

বিকল্প (1) মিথ্যা

(3): f(x) = ex cos\(\rm\frac{1}{x}\)

\(\lim_{x\to0}\)ex cos\(\rm\frac{1}{x}\) বিদ্যমান নেই তাই f(x) = ex cos\(\rm\frac{1}{x}\) (0, 1) এ অভিন্নভাবে অবিচ্ছিন্ন নয়

বিকল্প (3) মিথ্যা

(4): f(x) = cos x cos\(\rm\frac{\pi}{x}\)

\(\lim_{x\to0}\)cos x cos\(\rm\frac{\pi}{x}\) বিদ্যমান নেই কারণ \(\lim_{x\to0}\)cos\(\rm\frac{\pi}{x}\) বিদ্যমান নেই তাই f(x) = cos x cos\(\rm\frac{\pi}{x}\) (0, 1) এ অভিন্নভাবে অবিচ্ছিন্ন নয়

বিকল্প (4) মিথ্যা

(2): f(x) = e−1/x2

(এখানে f(x) (0, 1) এ অবিচ্ছিন্ন এবং x = 0 এবং x = 1 এ সীমা বিদ্যমান

তাই f(x) = e−1/x2 (0, 1) এ অভিন্নভাবে অবিচ্ছিন্ন

বিকল্প (2) সঠিক

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