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If (f:\mathbb{R}\to\mathbb{R}) is given by (f(x)=\frac{1}{1+x^2}), what is the range of (f)?यदि (f:\mathbb{R}\to\mathbb{R}) को (f(x)=\frac{1}{1+x^2}) से दिया गया है, तो (f) का परास कौन सा है?
Answer and explanationउत्तर और व्याख्या
Correct answer:सही उत्तर: ((0,1])((0,1])
Step 1: Since (x^2\geq 0), (1+x^2\geq 1). Step 2: The maximum value is (1) at (x=0), and the function approaches (0) but never reaches it. Step 3: For rational ranges, watch endpoint inclusion carefully.चरण 1: (x^2\geq 0), इसलिए (1+x^2\geq 1)। चरण 2: सबसे बड़ा मान (x=0) पर (1) है और मान (0) तक केवल पास जाता है, पहुँचता नहीं। चरण 3: भिन्नात्मक परास में सीमा जैसा व्यवहार ध्यान से देखें।
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