Question

lim x tends to 0 (1 -tanx)/(x-pi /4)​

Answer 1

Explanation:

$\circ \: \tt \: lim_{x \rightarrow0} \: ( \dfrac{1 - tan(x)}{x - \dfrac{ \pi}{4} } )\green \bigstar$

$\displaystyle \: \longrightarrow \: \tt \: ( \dfrac{1 - tan(0)}{0 - \dfrac{ \pi}{4} } )$

$\displaystyle \: \longrightarrow \: \tt \: ( \dfrac{1 - 0}{0 - \dfrac{ \pi}{4} })$

$\displaystyle \: \longrightarrow \: \tt \: ( \dfrac{1 }{- \dfrac{ \pi}{4} })$

$\displaystyle \: \longrightarrow \: \tt \: - \dfrac{4}{ \pi} \: \red \bigstar$

[ Note: Refer to the attachment for some important limits formula's.]

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Know to more:

$\begin{gathered}\boxed{\boxed{\begin{minipage}{4cm}\displaystyle\circ\sf\:\int{1\:dx}=x+c\\\\\circ\sf\:\int{a\:dx}=ax+c\\\\\circ\sf\:\int{x^n\:dx}=\dfrac{x^{n+1}}{n+1}+c\\\\\circ\sf\:\int{sin\:x\:dx}=-cos\:x+c\\\\\circ\sf\:\int{cos\:x\:dx}=sin\:x+c\\\\\circ\sf\:\int{sec^2x\:dx}=tan\:x+c\\\\\circ\sf\:\int{e^x\:dx}=e^x+c\end{minipage}}}\end{gathered}$

Answer 2

Solution :-

$\implies\sf{lim_{x\to0}( \dfrac{1 - tan(x)}{x - \dfrac{\pi}{4} }) }$

$\implies\sf{( \dfrac{1 - tan(0)}{0 - \dfrac{\pi}{4} }) }$

$\implies\sf( \dfrac{1 - 0}{0 - \dfrac{\pi}{4}} )$

$\implies\sf( \dfrac{1}{ - \dfrac{\pi}{4}} )$

$\implies{\boxed{\red{\bf{ - \dfrac{ 4}{\pi} }}}}$