Question

lim x=π/2 tan2x/x-π/2​

Answer 1

Answer:

Step-by-step explanation:

Answer 2

Solution :

$\sf \underset{x \longrightarrow \: \frac{ \pi}{2} }{lim} \: \: \frac{tan2x}{x - \frac{ \pi}{2} }$

Let h = x - π/2

$\sf \underset{h\longrightarrow \: 0 }{lim} \: \: \frac{tan2x}{h} \\ \\ \sf \underset{h \longrightarrow \: 0 }{lim} \: \frac{ tan \: 2(h + \frac{ \pi}{2} )}{h} \\ \\ \sf \underset{h \longrightarrow \: 0 }{lim} \: \: \frac{tan \: (2h + \pi)}{h} \\ \\ \sf \underset{h \longrightarrow \: 0 }{lim} \: \: \frac{tan \: ( \pi + 2h)}{h} \\ \\ \sf \underset{h\longrightarrow \: 0}{lim} \: \: \frac{tan2h}{h} \\ \\ \sf \underset{h\longrightarrow \: 0 }{lim} \: \: \frac{tan \: 2h}{h \times \frac{2}{2} } \\ \\ \sf \underset{h\longrightarrow \: 0 }{lim} \: \: \frac{tan2h}{2h \times \frac{1}{2} } \\ \\ \red{\boxed{ \sf\underset{h\longrightarrow \: 0 }{lim} \: \: \frac{tan \: x}{x} = 1}} \\ \\ = 1 \times 2 \\ \\ = 2$

━━━━━━━━━━━━━━━━━━━━━━━━━

⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀⠀Done࿐