If [ _x ( 1 + a x - 4 x^2 )^2x = e^3,] then 'a' is equal to :

Question

If    $$\mathop {\lim }\limits_{x \to \infty } {\left( {1 + {a \over x} - {4 \over {{x^2}}}} \right)^{2x}} = {e^3},$$ then 'a' is equal to :

Accepted Answer

Given,

$$\mathop {\lim }\limits_{x \to  \propto } {\left( {1 + {a \over x} - {4 \over {{x^2}}}} \right)^{2x}} = {e^3}$$

So,   $$\mathop {\lim }\limits_{x \to  \propto } {\left( {1 + {a \over x} - {4 \over {{x^2}}}} \right)^{2x}}\,\left[ {{1^ \propto }\,\,\,form} \right]$$

$$ = {e^{\mathop {\lim }\limits_{x \to  \propto } \left[ {\left( {1 + {a \over x} - {4 \over {{x^2}}} - 1} \right)2x} \right]}}$$

$$ = {e^{\mathop {\lim }\limits_{x \to  \propto } \left( {2a - {8 \over x}} \right)}}$$

$= {e^{2a}}$

$$ \therefore $$   e2a = e3

$$ \therefore $$    2a = 3

$$ \Rightarrow $$   a $=$ $${3 \over 2}$$

Solutions · 1