The value of lim x - 0tanx7-cos3 x714-x is

L=lim _x → 0(tanx7+cos3 x7)^14/x nbsp; (1^∞ form) =exp{lim _x → 014x(tanx7+cos3 x7 1)} =exp{2lim _x → 0(tanx7x7+3(cos3 x7 13 x7))} =exp{2lim _x → 0(tanx7x7 ...

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Standard XII

Mathematics

L'Hospital Rule to Remove Indeterminate Form

The value of ...

Question

The value of $lim x \to 0 {(tan \frac{x}{7} + cos \frac{3 x}{7})}^{14 / x}$ is

e^{- 1}

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e^{2}

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1

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e

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Solution

The correct option is B $e^{2}$
$L = lim x \to 0 {(tan \frac{x}{7} + cos \frac{3 x}{7})}^{14 / x}$ $(1^{\infty}$ form $)$

$= exp {lim x \to 0 \frac{14}{x} (tan \frac{x}{7} + cos \frac{3 x}{7} - 1)} = exp ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ 2 lim x \to 0 ⎛ ⎜ ⎜ ⎜ ⎝ \frac{tan \frac{x}{7}}{\frac{x}{7}} + 3 ⎛ ⎜ ⎜ ⎜ ⎝ \frac{cos \frac{3 x}{7} - 1}{\frac{3 x}{7}} ⎞ ⎟ ⎟ ⎟ ⎠ ⎞ ⎟ ⎟ ⎟ ⎠ ⎫ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎭$

$= exp ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ 2 lim x \to 0 ⎛ ⎜ ⎜ ⎜ ⎝ \frac{tan \frac{x}{7}}{\frac{x}{7}} - 3 ⎛ ⎜ ⎜ ⎜ ⎝ \frac{2 {sin}^{2} \frac{3 x}{14}}{\frac{3 x}{7}} ⎞ ⎟ ⎟ ⎟ ⎠ ⎞ ⎟ ⎟ ⎟ ⎠ ⎫ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎭$

$= exp ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ 2 lim x \to 0 ⎛ ⎜ ⎜ ⎜ ⎝ \frac{tan \frac{x}{7}}{\frac{x}{7}} - 3 ⎛ ⎜ ⎜ ⎜ ⎝ \frac{2 sin \frac{3 x}{14}}{\frac{3 x}{14} \cdot 2} \cdot sin \frac{3 x}{14} ⎞ ⎟ ⎟ ⎟ ⎠ ⎞ ⎟ ⎟ ⎟ ⎠ ⎫ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎭$

$= exp (2 (1 - 0))$
$= e^{2}$

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The value of limx→0(tanx7+cos3x7)14/x is

The value of $lim x \to 0 {(tan \frac{x}{7} + cos \frac{3 x}{7})}^{14 / x}$ is