help-please-lim-x-4-cos-4-x-tan-x-1-sin-4-x-

Question Number 196107 by maths_plus last updated on 18/Aug/23

$$ \\ $$$$\mathrm{help},\:\mathrm{please}\:! \\ $$$$\underset{{x}\:\rightarrow\:\frac{\boldsymbol{\pi}}{\mathrm{4}}} {{lim}}\:\frac{{cos}\left(\frac{\boldsymbol{\pi}}{\mathrm{4}}\:−{x}\right)−{tan}\:{x}}{\mathrm{1}−{sin}\left(\frac{\boldsymbol{\pi}}{\mathrm{4}}+{x}\right)}\:=\:???\: \\ $$

Answered by MM42 last updated on 21/Aug/23

$${hop}\rightarrow{lim}_{{x}\rightarrow\frac{\pi}{\mathrm{4}}} \:\frac{{sin}\left(\frac{\pi}{\mathrm{4}}−{x}\right)−\left(\mathrm{1}+{tan}^{\mathrm{2}} {x}\right)}{−{cos}\left(\frac{\pi}{\mathrm{4}}+{x}\right)} \\ $$$$=\frac{−\mathrm{2}}{\mathrm{0}}\begin{cases}{{x}\rightarrow\frac{\pi^{+} }{\mathrm{4}}\:\:\:\:\:\:=+\infty}\\{{x}\rightarrow\frac{\pi^{−} }{\mathrm{4}}\:\:\:\:\:\:=−\infty}\end{cases} \\ $$$$\Rightarrow\:{limit}\:\:{not}\:\:{exist} \\ $$$$ \\ $$

Commented by maths_plus last updated on 18/Aug/23

$$\mathrm{thank}\:\mathrm{you} \\ $$

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