lim_x→ #160; π2 (sec x tan x) ⇒lim_x→π2 [ 1cos x sin xcos x ] #160; ⇒lim_x→π2 [ 1 sin xcos x ] #160; ⇒lim_x→π/2(cos x x2 sin x2)^2c ...

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Substitution Method to Remove Indeterminate Form

limx →π2 secx...

Question

$lim x \to \frac{π}{2} (s e c x - t a n x)$

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Solution

${lim}_{x \to \frac{π}{2}} (s e c x - t a n x)$

$\Rightarrow {lim}_{x \to \frac{π}{2}} [\frac{1}{c o s x} - \frac{s i n x}{c o s x}]$

$\Rightarrow {lim}_{x \to \frac{π}{2}} [\frac{1 - s i n x}{c o s x}]$

$\Rightarrow {lim}_{x \to π / 2} \frac{(c o s x \frac{x}{2} - s i n \frac{x}{2})^{2}}{c o s^{2} \frac{x}{2} - s i n^{2} \frac{x}{2}}$

${∴1=sin2dfracx2+cos2x/2,sinx=2sinx/2cosx/2,cosx=cos2x2−sin2x2}$

$\Rightarrow {lim}_{x \to \frac{π}{2}} \frac{(c o s x / 2 - s i n x / 2)^{2}}{(c o s x / 2 - s i n x / 2) (c o s x / 2 + s i n x / 2)}$

$\Rightarrow {lim}_{x \to \frac{π}{2}} \frac{c o s x / 2 - s i n x / 2}{c o s x / 2 + s i n x / 2}$

$\Rightarrow 0$

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