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Question
Evaluate limy→0⎡⎢
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⎢⎣limx→∞exp[xℓn[1+ayx]]−exp[xℓn[1+byx]]y⎤⎥
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Evaluate limy→0⎡⎢
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⎢⎣limx→∞exp[xℓn[1+ayx]]−exp[xℓn[1+byx]]y⎤⎥
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Solution
The correct option is A a−b
limy→0⎡⎢
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⎢⎣limx→∞exln(1+ay)−exln(1+by)y⎤⎥
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Rewriting the above equation
⇒limy→0⎡⎢
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⎢⎣limx→∞eln(1+ayx)ayx×1ay−eln(1+byx)byx×1byy⎤⎥
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=limy→0{eay−ebyy}=limy→0[a(eay−1)ay−b(eby−1)by]⇒a−b
limy→0⎡⎢ ⎢ ⎢ ⎢⎣limx→∞exln(1+ay)−exln(1+by)y⎤⎥ ⎥ ⎥ ⎥⎦
Rewriting the above equation
⇒limy→0⎡⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢⎣limx→∞eln(1+ayx)ayx×1ay−eln(1+byx)byx×1byy⎤⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥⎦
=limy→0{eay−ebyy}=limy→0[a(eay−1)ay−b(eby−1)by]⇒a−b
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