Question

lim tan2x/x-π/2
x➡️π/2​

Answer 1

$\sf{\displaystyle \lim_{x \to \frac{\pi}{2}} \frac{tan 2x}{x - \frac{\pi}{2}} }$

On putting the value of π/2 in place of x, we find that the result obtained is not defined. Thus, we now mould the equation and then we'll put the value of π/2 in place of x, so as to get a definite answer.

let $\sf {x - \frac{\pi}{2} = h}$, then our limit will become:-

$\sf{\displaystyle \lim_{x \to \frac{\pi}{2}} = \displaystyle \lim_{x - \frac{\pi}{2} \to 0} = \displaystyle \lim_{h \to 0} }$

now we find the value of 2x, in terms of h:-

$\sf{h = x - \frac{\pi}{2} → h = \frac{2x-\pi}{2} → 2h = 2x-\pi → 2x = 2h + \pi}$

Our final expression becomes:

→ $\sf{\displaystyle \lim_{h \to 0} \frac{tan (2h + \pi)}{h} = \displaystyle \lim_{h \to 0} \frac{tan 2h}{h} }$

{Property used: tan (x + π) = tan x}

multiply 2 in the limit, on both sides; and in the expression, in both parts of fraction:-

→ $\sf{\displaystyle \lim_{2h \to 2(0)} \frac{2 tan 2h}{2h} }$

{Property used: $\bold{\sf{\displaystyle \lim_{x \to 0} \frac{tan x}{x}}}$ }

→ 2(1)

→ $\boxed{\sf {\red{2}}}$