(6) $$ \lim _{x ightarrow 1} \frac{ an \pi x}{x-1} $$

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(6) $$ \lim _{x ightarrow 1} \frac{	an \pi x}{x-1} $$

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Calculate the limit $$ \lim_{x ightarrow 1} \frac{	an(\pi x)}{x-1} $$.
Evaluate the limit by following steps:
- step0: Evaluate using L'Hopital's rule:
  $$\lim _{xightarrow 1}\left(\frac{	an\left(\pi xight)}{x-1}ight)$$
- step1: Use the L'Hopital's rule:
  $$\lim _{xightarrow 1}\left(\frac{\frac{d}{dx}\left(	an\left(\pi xight)ight)}{\frac{d}{dx}\left(x-1ight)}ight)$$
- step2: Find the derivative:
  $$\lim _{xightarrow 1}\left(\frac{\pi \sec^{2}\left(\pi xight)}{1}ight)$$
- step3: Simplify the expression:
  $$\lim _{xightarrow 1}\left(\pi \sec^{2}\left(\pi xight)ight)$$
- step4: Rewrite the expression:
  $$\pi 	imes \lim _{xightarrow 1}\left(\sec^{2}\left(\pi xight)ight)$$
- step5: Calculate:
  $$\pi 	imes 1$$
- step6: Multiply the numbers:
  $$\pi $$
The limit of $$ \frac{	an \pi x}{x-1} $$ as $$ x $$ approaches 1 is $$ \pi $$.
The limit is $$ \pi $$.

(6)

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