Evaluate the following limit : lim(x→π/3) (√3 - tan x)/(π - 3x)

Question

Evaluate the following limit : \(\lim\limits_{\text x \to\pi/3}\cfrac{\sqrt3-tan\,\text x}{\pi-3\text x} \)

lim(x→π/3) (√3 - tan x)/(π - 3x)

Answer 1

We have, \(\lim\limits_{\text x \to\pi/3}\cfrac{\sqrt3-tan\,\text x}{\pi-3\text x} \)

If x → \(\cfrac{\pi}3\), \(\cfrac{\pi}3-\text x\) → 0, π - 3x → 0

Let \(\cfrac{\pi}3\) - x = y then y →0

Hence, \(\lim\limits_{\text x \to\pi/3}\cfrac{\sqrt3-tan\,\text x}{\pi-3\text x} \) = \(\cfrac43\)