Calculating $ \lim_{x\to \frac {\pi}{6}} \left(\frac {\sin(x- \frac{\pi}{6})}{\frac{{\sqrt3}}{2}-\cos x}\right)$, without using L'Hospital rule

Question

Does anyone have any idea how to transform this to calculate the limit as x approaches $\frac {\pi}{6}$, without using L'Hospital rule?

$$ \lim_{x\to \Large{\frac {\pi}{6}}} \left(\frac {\sin(x- \frac{\pi}{6})}{\frac{{\sqrt3}}{2}-\cos x}\right)$$

Answer 1

Let put $t=x-\frac{\pi}{6}$

we will compute

$$\lim_{t\to 0}\frac{\sin(t)}{\frac{\sqrt{3}}{2}-\cos(t+\frac{\pi}{6})}=$$

$$\lim_{t\to0}\frac{\sin(t)}{ \frac{\sqrt{3}}{2}(1- \cos(t))+\frac{\sin(t)}{2} }=$$

$$\lim_{t\to 0}\frac{1}{ \frac{1}{2}+\frac{\sqrt{3}}{2}\frac{t^2(1-\cos(t))}{t^2\sin(t)} }=2.$$

Answer 2

Let $f(x) = \sin (x-\pi/6), g(x) = \cos x.$ The expression takes the form

$$-\frac{f(x) - f(\pi/6)}{g(x) - g(\pi/6)} = -\frac{(f(x) - f(\pi/6))/( x-\pi/6)}{(g(x) - g(\pi/6))/(x-\pi/6)}.$$

As $x\to \pi/6,$ the last expression $\to -f'(\pi/6)/g'(\pi/6)$ by the defintion of the derivative. That is easy to calculate, and no L'Hopital was used.

Answer 3

Using trig identities $$\sin 2\theta = 2 \sin\theta \cos\theta\qquad\text{and}\qquad \cos\alpha - \cos\beta = -2 \sin\frac{\alpha - \beta}{2}\sin\frac{\alpha + \beta}{2}$$ ... we can write ...

$$\begin{align} \sin\left( x - \frac{\pi}{6}\right) &= 2\sin\left(\frac{x}{2} - \frac{\pi}{12}\right) \cos\left(\frac{x}{2} - \frac{\pi}{12}\right) \tag{1} \\[8pt] \frac{\sqrt{3}}{2}-\cos x &= \cos\frac{\pi}{6} - \cos x \\[4pt] &= 2\sin\left(\frac{\pi}{12} - \frac{x}{2}\right) \sin\left(\frac{\pi}{12} + \frac{x}{2}\right) \\[8pt] &= -2\sin\left(\frac{x}{2} - \frac{\pi}{12}\right) \sin\left(\frac{x}{2} + \frac{\pi}{12}\right) \tag{2} \end{align}$$

... so that, for $\sin\left(\frac{x}{2} -\frac{\pi}{12}\right) \neq 0$ (that is, $x$ in a neighborhood of $\frac{\pi}{6}$ ), ...

$$\frac{\sin\left( x - \dfrac{\pi}{6}\right)}{\dfrac{\sqrt{3}}{2} - \cos x} = - \frac{\cos\left(\dfrac{x}{2}-\dfrac{\pi}{12}\right)}{\sin\left(\dfrac{x}{2} + \dfrac{\pi}{12}\right)} \tag{3}$$

Taking the limit on the right-hand side of $(3)$ amounts to substitution.