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Introduction to limits

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Question

$If \lim_{x \to 1} {(1 + ax + {bx}^{2})}^{\frac{c}{(x - 1)}} = e^{3},$ then the value of bc is ______

Moderate

Solution

$\begin{array}{l} \lim_{x \to 1} {(1 + ax + {bx}^{2})}^{\frac{c}{(x - 1)}} = e^{3} \\ or e^{\lim_{x \to 1} {(1 + ax + {bx}^{2} - 1)}^{\frac{c}{x - 1}}} = e^{3} \\ or e^{\lim_{x \to 1} \frac{c (ax + {bx}^{2})}{x - 1}} = e^{3} \\ or \lim_{h \to 0} \frac{c (ax + {bx}^{2})}{x - 1} = 3 \\ or \lim_{h \to 0} \frac{c (a (1 + h) + b {(a + h)}^{2})}{1 + h - 1} = 3 \\ or \lim_{h \to 0} \frac{(ca + b) + (ac + 2 b) h + {bh}^{2}}{h} = 3 \\ or ca + b = 0 and ac + 2 b = 3 \\ or b = 3 and ac = - 3 \\ Also, the form must be 1^{\infty} for which a + b = 0, i . e ., a = - 3 and c = 1 . \end{array}$

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