Lim x - 0sin 2 4 x2-x4

Lim_x →0sin^24x^2/x^4 =lim_x →0(sin4x^2)^2/x^4 =lim_x →0(sin4x^2)^2/(x^2)^2 = (lim_x →0sin4x^2/x^2 )^2 = (lim_4x^2 →0sin4x^2/4x^2 )× 16 =1 × 16=16 [ ∵li ...

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Byju's Answer

Standard XII

Mathematics

Standard Limits to Remove Indeterminate Form

lim x → 0sin ...

Question

${lim}_{x \to 0} \frac{s i n^{2} 4 x^{2}}{x^{4}}$

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Solution

${lim}_{x \to 0} \frac{s i n^{2} 4 x^{2}}{x^{4}}$

$= {lim}_{x \to 0} \frac{(s i n 4 x^{2})^{2}}{x^{4}}$

$= {lim}_{x \to 0} \frac{(s i n 4 x^{2})^{2}}{(x^{2})^{2}}$

$= {({lim}_{x \to 0} \frac{s i n 4 x^{2}}{x^{2}})}_{x \to 0}^{2}$

$= ({lim}_{4 x^{2} \to 0} \frac{s i n 4 x^{2}}{4 x^{2}}) \times 16$

$= 1 \times 16 = 16 [∵ {lim}_{x \to 0} \frac{s i n x}{x} = 1]$

=16

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limx→0sin24x2x4

limx→0sin24x2x4 =limx→0(sin4x2)2x4 =limx→0(sin4x2)2(x2)2 =(limx→0sin4x2x2)2 =(lim4x2→0sin4x24x2)×16 =1×16=16 [∵limx→0sinxx=1] =16

${lim}_{x \to 0} \frac{s i n^{2} 4 x^{2}}{x^{4}}$