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Question Number 280 by arnav last updated on 25/Jan/15
Evaluate lim_(x→π/4) ((cos x−sin x)/((π/4−x)(cos x+sin x)))
$$\mathrm{Evaluate}\:\underset{{x}\rightarrow\pi/\mathrm{4}} {\mathrm{lim}}\frac{\mathrm{cos}\:{x}−\mathrm{sin}\:{x}}{\left(\pi/\mathrm{4}−{x}\right)\left(\mathrm{cos}\:{x}+\mathrm{sin}\:{x}\right)} \\ $$
Answered by 123456 last updated on 18/Dec/14
lim_(x→π/4) ((cos x−sin x)/(((π/4)−x)(cos x+sin x)))→(0/0) =lim_(x→π/4) ((−sin x−cos x)/(−(cos x+sin x)+((π/4)−x)(−sin x+cos x))) =((−((√2)/2)−((√2)/2))/(−(((√2)/2)+((√2)/2))+((π/4)−(π/4))(−((√2)/2)+((√2)/2)))) =1
$$\underset{{x}\rightarrow\pi/\mathrm{4}} {\mathrm{lim}}\frac{\mathrm{cos}\:{x}−\mathrm{sin}\:{x}}{\left(\frac{\pi}{\mathrm{4}}−{x}\right)\left(\mathrm{cos}\:{x}+\mathrm{sin}\:{x}\right)}\rightarrow\frac{\mathrm{0}}{\mathrm{0}} \\ $$$$=\underset{{x}\rightarrow\pi/\mathrm{4}} {\mathrm{lim}}\frac{−\mathrm{sin}\:{x}−\mathrm{cos}\:{x}}{−\left(\mathrm{cos}\:{x}+\mathrm{sin}\:{x}\right)+\left(\frac{\pi}{\mathrm{4}}−{x}\right)\left(−\mathrm{sin}\:{x}+\mathrm{cos}\:{x}\right)} \\ $$$$=\frac{−\frac{\sqrt{\mathrm{2}}}{\mathrm{2}}−\frac{\sqrt{\mathrm{2}}}{\mathrm{2}}}{−\left(\frac{\sqrt{\mathrm{2}}}{\mathrm{2}}+\frac{\sqrt{\mathrm{2}}}{\mathrm{2}}\right)+\left(\frac{\pi}{\mathrm{4}}−\frac{\pi}{\mathrm{4}}\right)\left(−\frac{\sqrt{\mathrm{2}}}{\mathrm{2}}+\frac{\sqrt{\mathrm{2}}}{\mathrm{2}}\right)} \\ $$$$=\mathrm{1} \\ $$
Answered by 123456 last updated on 22/Dec/14
=lim_(x→π/4) ((cos x−sin x)/(((π/4)−x)(cos x+sin x))) =lim_(x→π/4) ((sin ((π/4)−x))/(((π/4)−x)sin ((π/4)+x)))∙((√2)/( (√2))) =lim_(x→π/4) ((sin ((π/4)−x))/((π/4)−x))lim_(x→π/4) (1/(sin ((π/4)+x))) =lim_(y→0) ((sin y)/y)∙(1/(sin ((π/4)+(π/4)))) =1∙(1/(sin (π/2)))=1
$$=\underset{{x}\rightarrow\pi/\mathrm{4}} {\mathrm{lim}}\frac{\mathrm{cos}\:{x}−\mathrm{sin}\:{x}}{\left(\frac{\pi}{\mathrm{4}}−{x}\right)\left(\mathrm{cos}\:{x}+\mathrm{sin}\:{x}\right)} \\ $$$$=\underset{{x}\rightarrow\pi/\mathrm{4}} {\mathrm{lim}}\frac{\mathrm{sin}\:\left(\frac{\pi}{\mathrm{4}}−{x}\right)}{\left(\frac{\pi}{\mathrm{4}}−{x}\right)\mathrm{sin}\:\left(\frac{\pi}{\mathrm{4}}+{x}\right)}\centerdot\frac{\sqrt{\mathrm{2}}}{\:\sqrt{\mathrm{2}}} \\ $$$$=\underset{{x}\rightarrow\pi/\mathrm{4}} {\mathrm{lim}}\frac{\mathrm{sin}\:\left(\frac{\pi}{\mathrm{4}}−{x}\right)}{\frac{\pi}{\mathrm{4}}−{x}}\underset{{x}\rightarrow\pi/\mathrm{4}} {\mathrm{lim}}\frac{\mathrm{1}}{\mathrm{sin}\:\left(\frac{\pi}{\mathrm{4}}+{x}\right)} \\ $$$$=\underset{{y}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\mathrm{sin}\:{y}}{{y}}\centerdot\frac{\mathrm{1}}{\mathrm{sin}\:\left(\frac{\pi}{\mathrm{4}}+\frac{\pi}{\mathrm{4}}\right)} \\ $$$$=\mathrm{1}\centerdot\frac{\mathrm{1}}{\mathrm{sin}\:\frac{\pi}{\mathrm{2}}}=\mathrm{1} \\ $$

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