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Question Number 203247 by mathlove last updated on 13/Jan/24

lim_(x→0) x tan(π/2)(1+x)=?

$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\:{x}\:{tan}\frac{\pi}{\mathrm{2}}\left(\mathrm{1}+{x}\right)=? \\ $$

Answered by MM42 last updated on 13/Jan/24

=lim_(x→0) −xcot(π/2)x=lim_(x→0) −(((π/2)xcos(π/2)x)/(sin(π/2)x))×(2/π) = −(2/π) ✓

$$={lim}_{{x}\rightarrow\mathrm{0}} \:−{xcot}\frac{\pi}{\mathrm{2}}{x}={lim}_{{x}\rightarrow\mathrm{0}} \:−\frac{\frac{\pi}{\mathrm{2}}{xcos}\frac{\pi}{\mathrm{2}}{x}}{{sin}\frac{\pi}{\mathrm{2}}{x}}×\frac{\mathrm{2}}{\pi} \\ $$$$=\:−\frac{\mathrm{2}}{\pi}\:\checkmark \\ $$$$ \\ $$

Commented by mathlove last updated on 13/Jan/24

way tan(π/2)(1+x)=−cotx

$${way}\:{tan}\frac{\pi}{\mathrm{2}}\left(\mathrm{1}+{x}\right)=−{cotx} \\ $$

Commented by MM42 last updated on 13/Jan/24

ok −cot(π/2)x

$${ok} \\ $$$$−{cot}\frac{\pi}{\mathrm{2}}{x} \\ $$

Answered by mr W last updated on 13/Jan/24

lim_(x→0) x tan (π/2)(1+x) =lim_(x→0) ((x sin ((π/2)+((πx)/2)))/(cos ((π/2)+((πx)/2)))) =lim_(x→0) ((x sin ((π/2)−((πx)/2)))/(−cos ((π/2)−((πx)/2)))) =lim_(x→0) ((x cos (((πx)/2)))/(−sin (((πx)/2)))) =−(2/π)lim_(x→0) (((((πx)/2)))/(sin (((πx)/2))))×cos (((πx)/2)) =−(2/π)

$$\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}{x}\:\mathrm{tan}\:\frac{\pi}{\mathrm{2}}\left(\mathrm{1}+{x}\right) \\ $$$$=\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{{x}\:\mathrm{sin}\:\left(\frac{\pi}{\mathrm{2}}+\frac{\pi{x}}{\mathrm{2}}\right)}{\mathrm{cos}\:\left(\frac{\pi}{\mathrm{2}}+\frac{\pi{x}}{\mathrm{2}}\right)} \\ $$$$=\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{{x}\:\mathrm{sin}\:\left(\frac{\pi}{\mathrm{2}}−\frac{\pi{x}}{\mathrm{2}}\right)}{−\mathrm{cos}\:\left(\frac{\pi}{\mathrm{2}}−\frac{\pi{x}}{\mathrm{2}}\right)} \\ $$$$=\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{{x}\:\mathrm{cos}\:\left(\frac{\pi{x}}{\mathrm{2}}\right)}{−\mathrm{sin}\:\left(\frac{\pi{x}}{\mathrm{2}}\right)} \\ $$$$=−\frac{\mathrm{2}}{\pi}\underset{{x}\rightarrow\mathrm{0}} {\mathrm{lim}}\frac{\left(\frac{\pi{x}}{\mathrm{2}}\right)}{\mathrm{sin}\:\left(\frac{\pi{x}}{\mathrm{2}}\right)}×\mathrm{cos}\:\left(\frac{\pi{x}}{\mathrm{2}}\right) \\ $$$$=−\frac{\mathrm{2}}{\pi} \\ $$

Commented by mathlove last updated on 13/Jan/24

thats right thanks sir

$${thats}\:{right} \\ $$$${thanks}\:{sir} \\ $$

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