Lim x -42-cosec2 x-1-cotx

Lim_x→π/42 cosec^2x/1 cotx =lim_x→π/42 (1+cot^2x)/1 cotx =lim_x→π/42 1 cot^2x/1 cotx =lim_x→π/41 cot^2x/1 cotx =lim_x→π/4(1 cotx)(1+cotx)/1 cotx =lim_x→(1+cotx) ...

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Byju's Answer

Standard XII

Mathematics

Expansions to Remove Indeterminate Form

lim x →π/42-c...

Question

${lim}_{x \to \frac{π}{4}} \frac{2 - c o s e c^{2} x}{1 - c o t x}$

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Solution

${lim}_{x \to \frac{π}{4}} \frac{2 - c o s e c^{2} x}{1 - c o t x}$

$= {lim}_{x \to \frac{π}{4}} \frac{2 - (1 + c o t^{2} x)}{1 - c o t x}$

$= {lim}_{x \to \frac{π}{4}} \frac{2 - 1 - c o t^{2} x}{1 - c o t x}$

$= {lim}_{x \to \frac{π}{4}} \frac{1 - c o t^{2} x}{1 - c o t x}$

$= {lim}_{x \to \frac{π}{4}} \frac{(1 - c o t x) (1 + c o t x)}{1 - c o t x}$

$= {lim}_{x \to} (1 + c o t x)$

$= 1 + c o t \frac{π}{4}$

=1+1=2

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limx→π42−cosec2x1−cotx

limx→π42−cosec2x1−cotx =limx→π42−(1+cot2x)1−cotx =limx→π42−1−cot2x1−cotx =limx→π41−cot2x1−cotx =limx→π4(1−cotx)(1+cotx)1−cotx =limx→(1+cotx) =1+cotπ4 =1+1=2

${lim}_{x \to \frac{π}{4}} \frac{2 - c o s e c^{2} x}{1 - c o t x}$