Question
Download Solution PDF\(\lim\limits_{x \to 4} \frac{3-\sqrt{5+x}}{x-4}\) किसके बराबर है?
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFसंकल्पना:
L-हॉस्पिटल नियम: माना f(x) और g(x) दो फलन हैं
मान लीजिए कि हमारे पास निम्नलिखित मामलों में से एक है,
I. \(\lim\limits_{x \to a} \frac{f(x)}{g(x)} =\frac{0}{0}\)
II. \(\lim\limits_{x \to a} \frac{f(x)}{g(x)} =\frac{\infty }{\infty }\)
तब हम L-हॉस्पिटल नियम ⇔ \(\lim\limits_{x \to a} \frac{f(x)}{g(x)} =\lim\limits_{x \to a} \frac{f'(x)}{g'(x)}\)
नोट: हमें अंश और हर दोनों को x के सापेक्ष तब तक अवकलन करना होगा जब तक कि
\(\lim\limits_{x \to a} \frac{f(x)}{g(x)} =l\neq \frac{0}{0}\) जहां l एक परिमित मान है।
हल:
हमें \(\lim\limits_{x \to 4} \frac{3-\sqrt{5+x}}{x-4}\) का मान ज्ञात करना होगा
\(\lim\limits_{x \to 4} \frac{3-\sqrt{5+x}}{x-4}\) सीमा का रूप (0/0) है
L-हॉस्पिटल नियम लागू करना,
\(=\lim\limits_{x \to 4} \frac{0-\frac{1}{2\sqrt{5+x}}}{1-0}\)
\(=\frac{-1}{2\sqrt{9}}\)
\(=\frac{-1}{6}\)
सही विकल्प (2) है
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