Lim n -sina-2 n-sinh-2 n

= lim_n →∞sin (a/2^n)/sin ( b/2^n) = n →∞, Let 1/n⇒ h → 0 = lim_h → 0(a/2^1/h)/lim_h → 0(b/2^1/h) = ( lim_h → 0sin a/2^1/h/a/2^1/h×a/2^1/h )/ ( lim_h → 0sin b/2 ...

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Byju's Answer

Standard XII

Mathematics

Standard Limits to Remove Indeterminate Form

lim n →∞sina/...

Question

${lim}_{n \to \infty} \frac{s i n (\frac{a}{2^{n}})}{s i n (\frac{b}{2^{n}})}$

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Solution

$= {lim}_{n \to \infty} \frac{s i n (\frac{a}{2^{n}})}{s i n (\frac{b}{2^{n}})}$

$= n \to \infty$ , Let $\frac{1}{n} \Rightarrow h \to 0$

$= \frac{{lim}_{h \to 0} ⎛ ⎜ ⎝ \frac{a}{2^{\frac{1}{h}}} ⎞ ⎟ ⎠}{{lim}_{h \to 0} ⎛ ⎜ ⎝ \frac{b}{2^{\frac{1}{h}}} ⎞ ⎟ ⎠}$

$= \frac{⎛ ⎜ ⎜ ⎜ ⎜ ⎝ {lim}_{h \to 0} \frac{s i n \frac{a}{2^{\frac{1}{h}}}}{\frac{a}{2^{\frac{1}{h}}}} \times \frac{a}{2^{\frac{1}{h}}} ⎞ ⎟ ⎟ ⎟ ⎟ ⎠}{⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ {lim}_{h \to 0} \frac{s i n \frac{b}{2^{\frac{1}{h}}}}{\frac{b}{2^{\frac{1}{h}}}} \times \frac{b}{2^{\frac{1}{h}}} ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠}$

$= \frac{1 \times a}{1 \times b} = \frac{a}{b}$

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limn→∞sin(a2n)sin(b2n)

=limn→∞sin(a2n)sin(b2n) =n→∞, Let 1n⇒h→0 =limh→0⎛⎜⎝a21h⎞⎟⎠limh→0⎛⎜⎝b21h⎞⎟⎠ =⎛⎜ ⎜ ⎜ ⎜⎝limh→0sina21ha21h×a21h⎞⎟ ⎟ ⎟ ⎟⎠⎛⎜ ⎜ ⎜ ⎜ ⎜⎝limh→0sinb21hb21h×b21h⎞⎟ ⎟ ⎟ ⎟ ⎟⎠ =1×a1×b=ab

${lim}_{n \to \infty} \frac{s i n (\frac{a}{2^{n}})}{s i n (\frac{b}{2^{n}})}$