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JEE Mains · Maths · STD 12 - 5. continuity and differentiation
Let \(f :(0, \infty) \rightarrow(0, \infty)\) be a differentiable function such that \(f(1)= e\) and \(\lim \limits_{t \rightarrow x} \frac{t^{2} f^{2}(x)-x^{2} f^{2}(t)}{t-x}=0\) If \(f ( x )=1,\) then \(x\) is equal to
- A \(2e\)
- B \(\frac{1}{2 e }\)
- C \(e\)
- D \(\frac{1}{ e }\)
Answer & Solution
Correct Answer
(D) \(\frac{1}{ e }\)
Step-by-step Solution
Detailed explanation
\(L=\lim _{t \rightarrow x} \frac{t^{2} f^{2}(x)-x^{2} f^{2}(t)}{t-x}\) using L.H. rule \(L =\lim _{t \rightarrow x} \frac{2 tf ^{2}( x )- x ^{2} \cdot 2 f ^{\prime}( t ) \cdot f ( t )}{1}\) \(\Rightarrow L =2 xf ( x )\left( f ( x )- x f ^{\prime}( x )\right)=0\) (given)…
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