गणित | अध्याय - 5 | सांतत्य तथा अवकलनीयता

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यदि \\( f(x) = \\frac{1 - \\cos x}{x^2} \\), \\( x = 0 \\) पर सतत है, तो \\( f(0) \\) बराबर होगा -

\r\n\r\n ","optiona":" 1/2","optionb":" 2","optionc":" 1/4","optiond":" 4","rightans":"A","details":"\r\n\r\n\r\n \r\n सतत फलन\r\n \r\n \r\n\r\n\r\n

दिया गया फलन \\( x = 0 \\) पर सतत है इसलिए

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\r\n \\[\r\n \\lim_{h \\to 0} \\frac{2 \\sin^2 \\left(\\frac{h}{2} \\right)}{h^2}\r\n = \\frac{1}{2} \\lim_{h \\to 0} \\left( \\frac{\\sin \\left( \\frac{h}{2} \\right)}{\\frac{h}{2}} \\right)^2\r\n \\]\r\n

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\r\n \\[\r\n = \\frac{1}{2} \\times 1 = \\frac{1}{2}\r\n \\]\r\n

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\r\n \\( f(0) = \\frac{1}{2} \\)\r\n

"},{"id":14251,"question":"2. यदिफलन $\"\"$ x= 4 पर संतत है, तो kकामान है–","optiona":"4","optionb":"–4","optionc":"8","optiond":"–8","rightans":"A","details":"\r\n\r\n\r\n \r\n सीमा (Limit) का मान\r\n \r\n\r\n\r\n

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x = 4 पर दायाँ सीमा (R.H.L.) का मान निकालने पर:

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\r\n = lim f(x) = lim f(4 + h) = lim ((4 + h)² − 16) / (4 + h − 4)
\r\n h → 4⁺ h → 0 h → 0\r\n

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\r\n = f(4) = lim f(4 + h) = 8\r\n

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\r\n k = 8\r\n

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\r\n अतः सही विकल्प (C) है।\r\n

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"}];//]]>