The value of _ x 0 ( ax)^ cosec ^2 bx is- - Vidyadip

MCQ

The value of $\mathop {\lim }\limits_{x \to 0} {(\cos ax)^{cosec{^2}\ bx}}$ is-

A
${e^{\left( {\frac{{ - 8{b^2}}}{{{a^2}}}} \right)}}$
B
${e^{\left( {\frac{{ - 8{a^2}}}{{{b^2}}}} \right)}}$
✓
${e^{\left( {\frac{{ - {a^2}}}{{2{b^2}}}} \right)}}$
D
${e^{\left( {\frac{{ - {b^2}}}{{2{a^2}}}} \right)}}$

✓

Answer

Correct option: C.

${e^{\left( {\frac{{ - {a^2}}}{{2{b^2}}}} \right)}}$

c
$\mathop {\lim }\limits_{x \to 0} {\left( {\cos ax} \right)^{\cos e{c^2}bx}}\left( {{1^\infty }form} \right)$

$ = {e^{\mathop {\lim }\limits_{x \to 0} }}\left( {\cos ax - 1} \right) \times \frac{1}{{{{\sin }^2}bx}}$

$ = {e^{\mathop {\lim }\limits_{x \to 0} }} - \frac{{1 - \cos ax}}{{{{\left( {ax} \right)}^2}}} \times \frac{{{{\left( {ax} \right)}^2}}}{{{{\left( {\frac{{\sin bx}}{{bx}} \times bx} \right)}^2}}}$

$ = {e^{ - \frac{1}{2}\frac{{{a^2}}}{{{b^2}}}}}$

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