Question
Download Solution PDFफलन f(x) = (x + 1)cotx x = 0 पर सतत रहेगा यदि f(0) का मान ________ है।
Answer (Detailed Solution Below)
Detailed Solution
Download Solution PDFसंकल्पना:
एक फलन को सतत कहा जाता है यदि
\(\rm \displaystyle \lim_{x\rightarrow a} f(x) = f(a)\)
गणना:
दिया गया:
f(x) = (x + 1)cotx
दोनों तरफ से log लेना
log (f(x)) = cot x log (x + 1)
x = 0 पर निरंतरता की जाँच के लिए
\(\rm \displaystyle \lim_{x\rightarrow 0} log(f(x)) = \rm \displaystyle \lim_{x\rightarrow 0} \cot x \ log(x\ +\ 1)\)
\(\rm \displaystyle \lim_{x\rightarrow 0} log(f(x)) = \rm \displaystyle \lim_{x\rightarrow 0} \frac { \ log(x\ +\ 1)}{tan \ x}\) = 0/0
L--हॉस्पिटल नियम का उपयोग करना
\(\rm \displaystyle \lim_{x\rightarrow 0} log(f(x)) = \rm \displaystyle \lim_{x\rightarrow 0} \frac { 1}{sec^2 \ x\ (x+1)} \)
\(= \rm \displaystyle \lim_{x\rightarrow 0} \frac {cos^2x }{(x+1)} \)
⇒ log (f(0)) = 1
⇒ f(0) = e1 = e
Last updated on May 6, 2025
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