The value of Ltx→0⁡21tan2⁡x+31tan2⁡x+41tan2⁡x+51tan2⁡x+61tan2⁡x3tan2⁡x=

Ltx→0⁡21tan2⁡x+31tan2⁡x+41tan2⁡x+51tan2⁡x+61tan2⁡x3tan2⁡x=Ltx→0⁡61tan2⁡x3tan2⁡x261tan2⁡x+361tan2⁡x+….+13tan2x =63(0+0+0....+1)0 =63 =216

The value of Ltx→0⁡21tan2⁡x+31tan2⁡x+41tan2⁡x+51tan2⁡x+61tan2⁡x3tan2⁡x=

Ltx→0⁡21tan2⁡x+31tan2⁡x+41tan2⁡x+51tan2⁡x+61tan2⁡x3tan2⁡x=Ltx→0⁡61tan2⁡x3tan2⁡x261tan2⁡x+361tan2⁡x+….+13tan2x =63(0+0+0....+1)0 =63 =216

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The value of ${Lt}_{x \to 0} {(2^{\frac{1}{\tan^{2} x}} + 3^{\frac{1}{\tan^{2} x}} + 4^{\frac{1}{\tan^{2} x}} + 5^{\frac{1}{\tan^{2} x}} + 6^{\frac{1}{\tan^{2} x}})}^{3 \tan^{2} x} =$

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216

answer is C.

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Detailed Solution

${Lt}_{x \to 0} {(2^{\frac{1}{\tan^{2} x}} + 3^{\frac{1}{\tan^{2} x}} + 4^{\frac{1}{\tan^{2} x}} + 5^{\frac{1}{\tan^{2} x}} + 6^{\frac{1}{\tan^{2} x}})}^{3 \tan^{2} x} =$

${Lt}_{x \to 0} {(6^{\frac{1}{\tan^{2} x}})}^{3 \tan^{2} x} {[{(\frac{2}{6})}^{\frac{1}{\tan^{2} x}} + {(\frac{3}{6})}^{\frac{1}{\tan^{2} x}} + \dots . + 1]}^{3 \tan^{2} x} = 6^{3} {(0 + 0 + 0 . . . . + 1)}^{0} = 6^{3} = 216$