AA's Limit Problem #?

Calculus Level 3

$\lim_{x \to \pi/3} \frac {2\sin (x\sec x) - \sqrt{1+\sec x}}{1+\cos x - \sqrt 3 \sin x} = \ ?$

The answer is 3.82645.

1 solution

Chew-Seong Cheong
Jul 22, 2020

$\begin{aligned} L & = \lim_{x \to \pi/3} \frac {2\sin (x \sec x) -\sqrt{1+\sec x}}{1+\cos x - \sqrt 3 \sin x} & \small \blue{\text{A 0/0 case, L'Hôpital's rule applies.}} \\ & = \lim_{x \to \pi/3} \frac {2\cos (x \sec x)(\sec x + x \sec x \tan x) - \frac {\sec x \tan x}{2\sqrt{1+\sec x}}}{-\sin x - \sqrt 3 \cos x} & \small \blue{\text{Differentiate up and down w.r.t. }x} \\ & = \frac {2 \cos \frac {2\pi}3\left(2+\frac \pi 3 \cdot 2 \cdot \sqrt 3\right)-1}{- \frac {\sqrt 3}2 - \frac {\sqrt 3}2} \\ & = \sqrt 3 + \frac {2\pi}3 \approx \boxed{3.83} \end{aligned}$

Reference: L'Hôpital's rule

@Aly Ahmed , I believe you can see the LaTex code in the problem question. All function must start with \ for example: \frac (for fraction), \sin x, \cos x, \tan x, \sec x, \lim ... You must leave a space between \sin and x, \cos and x, ... Use \ [ and \ ] instead of \ ( and \ ). Don't clamp the entire question in the title.

Chew-Seong Cheong - 10 months, 3 weeks ago

AA's Limit Problem #?

The answer is 3.82645.

1 solution

0 pending reports