Evaluate lim x -4sin x-cos x-x-4-4

Put (x π/4 )=y so that when x→π/4 then y→ 0. ∴x →π/4lim(sin x cos x)/(x π/4 ) =y→ 0lim{ sin(π/4 +y ) cos (π/4+y ) }/y [putting (x π/4)=y] =y→ 0lim{ ...

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Byju's Answer

Standard XI

Mathematics

Euler's Representation

Evaluate lim ...

Question

Evaluate $l i m x \to \frac{π}{4} \frac{(s i n x - c o s x)}{(x - \frac{π}{4})}$

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Solution

put $(x - \frac{π}{4}) = y$ so that when $x \to \frac{π}{4} t h e n y \to 0.$

$∴ l i m x \to \frac{π}{4} \frac{(s i n x - c o s x)}{(x - \frac{π}{4})}$

$= l i m y \to 0 \frac{{s i n (\frac{π}{4} + y) - c o s (\frac{π}{4} + y)}}{y} [p u t t i n g (x - \frac{π}{4}) = y]$

$= l i m y \to 0 \frac{{(\frac{π}{4} c o s y + c o s \frac{π}{4} s i n y) - (c o s \frac{π}{4} c o s y - s i n \frac{π}{4} s i n y)}}{y}$

$= \frac{2}{\sqrt{2}} \times l i m y \to 0 (\frac{s i n y}{y}) = (\sqrt{2} \times 1) = \sqrt{2} .$

Suggest Corrections

Evaluate limx→π4(sin x −cos x)(x−π4)

put (x−π4)=y so that when x→π4 then y→0. ∴limx→π4(sin x −cos x)(x−π4) =limy→0{sin(π4+y)−cos (π4+y)}y [putting (x−π4)=y] =limy→0{(π4cos y+cos π4 sin y)−(cos π4 cos y−sin π4 sin y)}y =2√2×limy→0(sin yy)=(√2×1)=√2.

Evaluate $l i m x \to \frac{π}{4} \frac{(s i n x - c o s x)}{(x - \frac{π}{4})}$