Try to solve it 1

$\large \lim_{x\to\frac13} \dfrac{\sqrt3 - \tan(\pi x)}{3x-1} = \ ?$

#Calculus #Limits

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Comments

We could just use L'Hopital's rule, but I'm assuming that you are looking for an approach that does not use this rule.

Let $u = x - \dfrac{1}{3}.$ Then $u \rightarrow 0$ as $x \rightarrow \dfrac{1}{3}$ and $x = u + \dfrac{1}{3},$ in which case

$\sqrt{3} - \tan(\pi x) = \sqrt{3} - \tan\left(\pi u + \dfrac{\pi}{3}\right) = \sqrt{3} - \dfrac{\tan(\pi u) + \tan(\frac{\pi}{3})}{1 - \tan(\pi u)\tan(\frac{\pi}{3})} =$

$\sqrt{3} - \dfrac{\tan(\pi u) + \sqrt{3}}{1 - \sqrt{3}\tan(\pi u)} = \dfrac{\sqrt{3} - 3\tan(\pi u) - \tan(\pi u) - \sqrt{3}}{1 - \sqrt{3}\tan(\pi u)} = \dfrac{-4\tan(\pi u)}{1 - \sqrt{3}\tan(\pi u)}.$

Thus the original limit can be written as

$\lim_{u \rightarrow 0} \dfrac{-4\tan(\pi u)}{3u*(1 - \sqrt{3}\tan(\pi u))} =$

$\lim_{u \rightarrow 0} \dfrac{-4\pi \tan(\pi u)}{3*(\pi u)} * \lim_{u \rightarrow 0} \dfrac{1}{1 - \sqrt{3}\tan(\pi u)} = -\dfrac{4\pi}{3} * \lim_{w \rightarrow 0} \dfrac{\tan(w)}{w} * 1 = \boxed{-\dfrac{4 \pi}{3}},$

where $w = \pi u \rightarrow 0$ as $u \rightarrow 0.$ Note also that $(1 - \sqrt{3}\tan(\pi u)) \rightarrow 1$ as $u \rightarrow 0.$

Brian Charlesworth - 5 years, 8 months ago

Nice! Approaches without using L'hopital's rule are always really good.

Kartik Sharma - 5 years, 8 months ago

Thanks! L'Hopital's is really useful if you want a quick result, but I like the challenge of trying to find an alternative approach. :)

Brian Charlesworth - 5 years, 8 months ago

@Brian Charlesworth – Another way is to put $\pi x = y$ and by first principles we can say the limit equals negative of the derivative of $\tan y$ at $y = \dfrac{ \pi}{3}$ . This is similar to L'Hospital but not completely.

Sudeep Salgia - 5 years, 8 months ago

$\frac{-4\pi}{3}$

Aditya Kumar - 5 years, 8 months ago

-4π/3

Jahid Rafi - 4 years, 11 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$